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Axiom ax-ac 9879
Description: Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC.

The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥. One entry in the pair is the member of 𝑥, and the other entry is some arbitrary member of that member of 𝑥. See the rewritten version ac3 9882 for a more detailed explanation. Theorem ac2 9881 shows an equivalent written compactly with restricted quantifiers.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 9885 is slightly shorter when the biconditional of ax-ac 9879 is expanded into implication and negation. In axac3 9884 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10101 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9912, ac5 9897, and ac7 9893. The Axiom of Regularity ax-reg 9053 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2b 9554. Equivalents to AC are the well-ordering theorem weth 9915 and Zorn's lemma zorn 9927. See ac4 9895 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 9053 for derivation of AC equivalents, we provide ax-ac2 9883 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9883 from ax-ac 9879 is shown by theorem axac2 9886, and the reverse derivation by axac 9887. Therefore, new proofs should normally use ax-ac2 9883 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Assertion
Ref Expression
ax-ac 𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡

Detailed syntax breakdown of Axiom ax-ac
StepHypRef Expression
1 vz . . . . . . 7 setvar 𝑧
2 vw . . . . . . 7 setvar 𝑤
31, 2wel 2116 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 2116 . . . . . 6 wff 𝑤𝑥
63, 5wa 399 . . . . 5 wff (𝑧𝑤𝑤𝑥)
7 vu . . . . . . . . . . . 12 setvar 𝑢
87, 2wel 2116 . . . . . . . . . . 11 wff 𝑢𝑤
9 vt . . . . . . . . . . . 12 setvar 𝑡
102, 9wel 2116 . . . . . . . . . . 11 wff 𝑤𝑡
118, 10wa 399 . . . . . . . . . 10 wff (𝑢𝑤𝑤𝑡)
127, 9wel 2116 . . . . . . . . . . 11 wff 𝑢𝑡
13 vy . . . . . . . . . . . 12 setvar 𝑦
149, 13wel 2116 . . . . . . . . . . 11 wff 𝑡𝑦
1512, 14wa 399 . . . . . . . . . 10 wff (𝑢𝑡𝑡𝑦)
1611, 15wa 399 . . . . . . . . 9 wff ((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
1716, 9wex 1781 . . . . . . . 8 wff 𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
18 vv . . . . . . . . 9 setvar 𝑣
197, 18weq 1965 . . . . . . . 8 wff 𝑢 = 𝑣
2017, 19wb 209 . . . . . . 7 wff (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2120, 7wal 1536 . . . . . 6 wff 𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2221, 18wex 1781 . . . . 5 wff 𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
236, 22wi 4 . . . 4 wff ((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2423, 2wal 1536 . . 3 wff 𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2524, 1wal 1536 . 2 wff 𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2625, 13wex 1781 1 wff 𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
Colors of variables: wff setvar class
This axiom is referenced by:  zfac  9880  ac2  9881
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