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With the axiom of dependent choice ( which is a weakened form of the axiom of choice ), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.
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With and axiom
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:
With this alternate notion of choice function, the axiom of choice can be compactly stated as
With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property which classical logic does not: whenever is proven constructively, then in fact is proven constructively for ( at least ) one particular, often called a witness.
With the full axiom of choice, every set is well-orderable, so every set has a cardinal ; we order the cardinals using the inherited ordering from the ordinal numbers.
With the axiom of choice we can well order the continuum ; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum.
With ordered pairs in hand, Class Comprehension enables defining relations and functions on sets as sets of ordered pairs, making possible the next axiom:
With and dependent
With the definitions used before 2012, the astronomical unit was dependent on the heliocentric gravitational constant, that is the product of the gravitational constant G and the solar mass M < sub >☉</ sub >.
With the breakup of the Soviet Union, the country, till that point highly dependent on exports to the USSR, had to make a radical shift in economic outlook: away from the East, and towards the West.
With Mahoré continuing to gravitate politically and economically toward France, and Comoros increasingly dependent on the French for help with its own considerable social, political, and economic problems, the issue of Mahoré diminished somewhat in urgency.
With most of the population remaining active in ( or indirectly dependent on ) the agricultural sector, the main pressure in most areas remained land clearing for crop and cattle farming.
With the exception of relatively recent, Asian-dominated investment in assembly firms along Honduras's northern coast, the country remains heavily dependent on United States-based multinational corporations for most of its investment needs in the early 1990s.
With the beginning of colonial rule in 1895, the Rift Valley and the surrounding Highlands became the enclave of white immigrants engaged in large-scale coffee farming dependent on mostly Kikuyu labor.
With this action, Asunción lost control of the Río de la Plata estuary and became dependent on Buenos Aires for maritime shipping.
With foreign revenue precariously dependent upon exports of cotton and aluminium, the economy is highly vulnerable to external shocks.
With all others in the British Empire, Antiguan slaves were emancipated in 1834, but remained economically dependent upon the plantation owners.
With the decline of the Roman Empire, distinctions appeared also in organization, since the bishops in the West were not dependent on the Emperor in Constantinople and did not come under the influence of the Caesaropapism in the Eastern Church.
With the abolition of the office of superintendent and of many other offices dependent upon it, the supreme control of the finances became vested in a royal council.
With this theory, the properties of a many-electron system can be determined by using functionals, i. e. functions of another function, which in this case is the spatially dependent electron density.
With such a large part of its economy dependent on tourism, the Aruban government is striving to increase business in other sectors to protect against possible industry slumps.
With the government already severely indebted to the tune of US $ 1 billion to the Soviet Union, Sukarno became increasingly dependent to Communist China for support.
With the administration dependent on Bavaria the Restoration and petty bourgeois mentality had quite a level playing field in Speyer.
With the rise of the Labor Party, the Free Trade Party had lost much of the middle ground to Barton and his followers, and it was increasingly dependent on conservatives, including militant Protestants.
With the ending of direct federal subsidies of the airport in the early 1990s, the Toronto Harbour Commission became dependent on subsidies from the City of Toronto.
With the Zeon backed into a corner, and the Federation dependent on Shiro and his team for victory, the star crossed lovers must decide where their true allegiances lie: with each other, or with their respective sides.
With marshlands largely destroyed, these cranes are increasingly dependent on wet paddy fields in India.
With the establishment of a Canadian subsidiary of Doubleday and Co., a firm which McClelland and Stewart had previously held Canadian distribution deals with, Jack started the move to a more Canadian-based catalogue: " I decided that I didn't want to be dependent on foreign agencies.
The other types of policies-“ Unit Linked ” and “ Unitised With Profits ” have a performance factor which is dependent directly on current investment market conditions.
With the decline of the Marcher Lords, and peace along the Welsh border, Cleobury's importance declined during the Middle Ages, leaving it as an undistinguished market town, with a long history, a market, and a dependent hinterland.
With seven of their 11 children still at home, she concentrated on her business affairs and on the education of those children still dependent on her.
With this action, the neosapiens became a viable race, no longer dependent on test tube breeding for survival.
With and choice
With the resumption of Soviet testing and their intransigence at the Geneva talks, however, the hope that this third choice would prove viable has been shaken.
With the exception of the Roman Catholic and the Orthodox Catholic Churches, most churches make no moral distinction between rhythm and mechanical or chemical contraceptives, allowing the couple free choice.
With a multi-million-dollar program finally on the very brink of success, Ford team officials faced a difficult choice.
With the success of muay Thai in the mixed martial arts, it has become the de facto style of choice for competitive stand-up fighters.
With PowerBooks being restricted to 68030s for several years, Macworld reviewers conceded that the best choice for power users was the PC-compatible Texas Instruments 80486 notebook, rather than the top-of-the-line PowerBook 180.
These programs resembled the " sympathetic " yet contradictory film Dances With Wolves of 1990, in which, according to Ella Shohat and Robert Stam, the narrative choice was to relate the Lakota story as told through a Euro-American voice, for wider impact among a general audience.
With our choice of units the marginal utility of the amount of the factor consumed directly by the optimizing consumer is again w, so the amount supplied of the factor too satisfies the condition of optimal allocation.
With no carrots left, Vladimir offers Estragon the choice between a turnip and a radish.
With no King, with no army ready, and King Edward's army at hand, the Scots had no choice.
With no other choice, Harvey accepts.
With no prospect of concrete assistance from Austria, and in view of the defeat of the Prussian – Russian coalition, which now sent peace signals to France, he felt he had no choice.
With Heathcliff now the master of both Wuthering Heights and Thrushcross Grange, Cathy has no choice but to leave Nelly and to go and live with Heathcliff and Hareton.
With this choice of a hyperplane, the points in the feature space that are mapped into the hyperplane are defined by the relation: Note that if becomes small as grows further away from, each element in the sum measures the degree of closeness of the test point to the corresponding data base point.
With the choice of defeat to the political left or dealing with the FN, the local RPR and UDF, to a minor national sensation, agreed to form an alliance with the FN, and together won the second round with 55 % of the vote.
With the first-overall choice, the Blue Jackets selected goaltender Rick Tabaracci from the Colorado Avalanche.
With sufficient growth, competitors enter the space, providing choice to the consumer, often competing merely on price as a cheaper alternative to the flagship brand's product offering.
With no choice, White re-established relations with the Croatans and tried to establish friendly relations with the tribes that Ralph Lane had battled the previous year.
With the universal success of " Good Vibrations ", Capitol Records had no choice but to back Wilson up for his next project, originally called Dumb Angel but soon re-titled SMiLE, which collaborator Van Dyke Parks would describe as a " teenage symphony to God ".
With constant law and order, made Jennings a good choice of living.
With the purchasing power to indulge any choice, Top Rung residents travel in style, both domestically and overseas.
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