Axiom ax-ac 9616

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The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 9619 for a more detailed explanation. Theorem ac2 9618 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 9622 is slightly shorter when the biconditional of ax-ac 9616 is expanded into implication and negation. In axac3 9621 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9838 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 9649, ac5 9634, and ac7 9630. The Axiom of Regularity ax-reg 8786 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2b 9286. Equivalents to AC are the well-ordering theorem weth 9652 and Zorn's lemma zorn 9664. See ac4 9632 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8786 for derivation of AC equivalents, we provide ax-ac2 9620 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9620 from ax-ac 9616 is shown by theorem axac2 9623, and the reverse derivation by axac 9624. Therefore, new proofs should normally use ax-ac2 9620 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Assertion

Ref	Expression
ax-ac	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡

Detailed syntax breakdown of Axiom ax-ac

Step	Hyp	Ref	Expression
1		vz	. . . . . . 7 setvar 𝑧
2		vw	. . . . . . 7 setvar 𝑤
3	1, 2	wel 2108	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 2108	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 386	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7		vu	. . . . . . . . . . . 12 setvar 𝑢
8	7, 2	wel 2108	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑤
9		vt	. . . . . . . . . . . 12 setvar 𝑡
10	2, 9	wel 2108	. . . . . . . . . . 11 wff 𝑤 ∈ 𝑡
11	8, 10	wa 386	. . . . . . . . . 10 wff (𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡)
12	7, 9	wel 2108	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑡
13		vy	. . . . . . . . . . . 12 setvar 𝑦
14	9, 13	wel 2108	. . . . . . . . . . 11 wff 𝑡 ∈ 𝑦
15	12, 14	wa 386	. . . . . . . . . 10 wff (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)
16	11, 15	wa 386	. . . . . . . . 9 wff ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
17	16, 9	wex 1823	. . . . . . . 8 wff ∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
18		vv	. . . . . . . . 9 setvar 𝑣
19	7, 18	weq 2005	. . . . . . . 8 wff 𝑢 = 𝑣
20	17, 19	wb 198	. . . . . . 7 wff (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
21	20, 7	wal 1599	. . . . . 6 wff ∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
22	21, 18	wex 1823	. . . . 5 wff ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
23	6, 22	wi 4	. . . 4 wff ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
24	23, 2	wal 1599	. . 3 wff ∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
25	24, 1	wal 1599	. 2 wff ∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
26	25, 13	wex 1823	1 wff ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))

This axiom is referenced by:  zfac  9617  ac2  9618