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Axiom ax-ac 9039
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The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 9042 for a more detailed explanation. Theorem ac2 9041 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 9045 is slightly shorter when the biconditional of ax-ac 9039 is expanded into implication and negation. In axac3 9044 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9257 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9072, ac5 9057, and ac7 9053. The Axiom of Regularity ax-reg 8255 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8711. Equivalents to AC are the well-ordering theorem weth 9075 and Zorn's lemma zorn 9087. See ac4 9055 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8255 for derivation of AC equivalents, we provide ax-ac2 9043 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9043 from ax-ac 9039 is shown by theorem axac2 9046, and the reverse derivation by axac 9047. Therefore, new proofs should normally use ax-ac2 9043 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 1939 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 1939 . . . . . 6 wff 𝑤𝑥
63, 5wa 382 . . . . 5 wff (𝑧𝑤𝑤𝑥)
7 vu . . . . . . . . . . . 12 setvar 𝑢
87, 2wel 1939 . . . . . . . . . . 11 wff 𝑢𝑤
9 vt . . . . . . . . . . . 12 setvar 𝑡
102, 9wel 1939 . . . . . . . . . . 11 wff 𝑤𝑡
118, 10wa 382 . . . . . . . . . 10 wff (𝑢𝑤𝑤𝑡)
127, 9wel 1939 . . . . . . . . . . 11 wff 𝑢𝑡
13 vy . . . . . . . . . . . 12 setvar 𝑦
149, 13wel 1939 . . . . . . . . . . 11 wff 𝑡𝑦
1512, 14wa 382 . . . . . . . . . 10 wff (𝑢𝑡𝑡𝑦)
1611, 15wa 382 . . . . . . . . 9 wff ((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
1716, 9wex 1694 . . . . . . . 8 wff 𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦))
18 vv . . . . . . . . 9 setvar 𝑣
197, 18weq 1824 . . . . . . . 8 wff 𝑢 = 𝑣
2017, 19wb 194 . . . . . . 7 wff (∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2120, 7wal 1472 . . . . . 6 wff 𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
2221, 18wex 1694 . . . . 5 wff 𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)
236, 22wi 4 . . . 4 wff ((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2423, 2wal 1472 . . 3 wff 𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2524, 1wal 1472 . 2 wff 𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
2625, 13wex 1694 1 wff 𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
Colors of variables: wff setvar class
This axiom is referenced by:  zfac  9040  ac2  9041
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