Axiom ax-ac 10473

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The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 10476 for a more detailed explanation. Theorem ac2 10475 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 10479 is slightly shorter when the biconditional of ax-ac 10473 is expanded into implication and negation. In axac3 10478 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10695 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 10506, ac5 10491, and ac7 10487. The Axiom of Regularity ax-reg 9606 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10145. Equivalents to AC are the well-ordering theorem weth 10509 and Zorn's lemma zorn 10521. See ac4 10489 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 9606 for derivation of AC equivalents, we provide ax-ac2 10477 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10477 from ax-ac 10473 is shown by Theorem axac2 10480, and the reverse derivation by axac 10481. Therefore, new proofs should normally use ax-ac2 10477 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 2109	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 2109	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 395	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7		vu	. . . . . . . . . . . 12 setvar 𝑢
8	7, 2	wel 2109	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑤
9		vt	. . . . . . . . . . . 12 setvar 𝑡
10	2, 9	wel 2109	. . . . . . . . . . 11 wff 𝑤 ∈ 𝑡
11	8, 10	wa 395	. . . . . . . . . 10 wff (𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡)
12	7, 9	wel 2109	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑡
13		vy	. . . . . . . . . . . 12 setvar 𝑦
14	9, 13	wel 2109	. . . . . . . . . . 11 wff 𝑡 ∈ 𝑦
15	12, 14	wa 395	. . . . . . . . . 10 wff (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)
16	11, 15	wa 395	. . . . . . . . 9 wff ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
17	16, 9	wex 1779	. . . . . . . 8 wff ∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
18		vv	. . . . . . . . 9 setvar 𝑣
19	7, 18	weq 1962	. . . . . . . 8 wff 𝑢 = 𝑣
20	17, 19	wb 206	. . . . . . 7 wff (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
21	20, 7	wal 1538	. . . . . 6 wff ∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
22	21, 18	wex 1779	. . . . 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 1538	. . 3 wff ∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
25	24, 1	wal 1538	. 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 1779	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  10474  ac2  10475