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Axiom ax-ac 9164
 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 The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 9167 for a more detailed explanation. Theorem ac2 9166 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 9170 is slightly shorter when the biconditional of ax-ac 9164 is expanded into implication and negation. In axac3 9169 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9382 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 9197, ac5 9182, and ac7 9178. The Axiom of Regularity ax-reg 8380 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8836. Equivalents to AC are the well-ordering theorem weth 9200 and Zorn's lemma zorn 9212. See ac4 9180 for comments about stronger versions of AC. In order to avoid uses of ax-reg 8380 for derivation of AC equivalents, we provide ax-ac2 9168 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9168 from ax-ac 9164 is shown by theorem axac2 9171, and the reverse derivation by axac 9172. Therefore, new proofs should normally use ax-ac2 9168 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 1978 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 1978 . . . . . 6 wff 𝑤𝑥
63, 5wa 383 . . . . 5 wff (𝑧𝑤𝑤𝑥)
7 vu . . . . . . . . . . . 12 setvar 𝑢
87, 2wel 1978 . . . . . . . . . . 11 wff 𝑢𝑤
9 vt . . . . . . . . . . . 12 setvar 𝑡
102, 9wel 1978 . . . . . . . . . . 11 wff 𝑤𝑡
118, 10wa 383 . . . . . . . . . 10 wff (𝑢𝑤𝑤𝑡)
127, 9wel 1978 . . . . . . . . . . 11 wff 𝑢𝑡
13 vy . . . . . . . . . . . 12 setvar 𝑦
149, 13wel 1978 . . . . . . . . . . 11 wff 𝑡𝑦
1512, 14wa 383 . . . . . . . . . 10 wff (𝑢𝑡𝑡𝑦)
1611, 15wa 383 . . . . . . . . 9 wff ((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
1716, 9wex 1695 . . . . . . . 8 wff 𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
18 vv . . . . . . . . 9 setvar 𝑣
197, 18weq 1861 . . . . . . . 8 wff 𝑢 = 𝑣
2017, 19wb 195 . . . . . . 7 wff (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2120, 7wal 1473 . . . . . 6 wff 𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2221, 18wex 1695 . . . . 5 wff 𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
236, 22wi 4 . . . 4 wff ((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2423, 2wal 1473 . . 3 wff 𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2524, 1wal 1473 . 2 wff 𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2625, 13wex 1695 1 wff 𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
 Colors of variables: wff setvar class This axiom is referenced by:  zfac  9165  ac2  9166
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