Axiom ax-ac 10461

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The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 10464 for a more detailed explanation. Theorem ac2 10463 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 10467 is slightly shorter when the biconditional of ax-ac 10461 is expanded into implication and negation. In axac3 10466 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10684 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 10494, ac5 10479, and ac7 10475. The Axiom of Regularity ax-reg 9564 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10133. Equivalents to AC are the well-ordering theorem weth 10497 and Zorn's lemma zorn 10509. See ac4 10477 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 9564 for derivation of AC equivalents, we provide ax-ac2 10465 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10465 from ax-ac 10461 is shown by Theorem axac2 10468, and the reverse derivation by axac 10469. Therefore, new proofs should normally use ax-ac2 10465 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 2147	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 2147	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 401	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7		vu	. . . . . . . . . . . 12 setvar 𝑢
8	7, 2	wel 2147	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑤
9		vt	. . . . . . . . . . . 12 setvar 𝑡
10	2, 9	wel 2147	. . . . . . . . . . 11 wff 𝑤 ∈ 𝑡
11	8, 10	wa 401	. . . . . . . . . 10 wff (𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡)
12	7, 9	wel 2147	. . . . . . . . . . 11 wff 𝑢 ∈ 𝑡
13		vy	. . . . . . . . . . . 12 setvar 𝑦
14	9, 13	wel 2147	. . . . . . . . . . 11 wff 𝑡 ∈ 𝑦
15	12, 14	wa 401	. . . . . . . . . 10 wff (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)
16	11, 15	wa 401	. . . . . . . . 9 wff ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
17	16, 9	wex 1812	. . . . . . . 8 wff ∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦))
18		vv	. . . . . . . . 9 setvar 𝑣
19	7, 18	weq 1995	. . . . . . . 8 wff 𝑢 = 𝑣
20	17, 19	wb 209	. . . . . . 7 wff (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
21	20, 7	wal 1568	. . . . . 6 wff ∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)
22	21, 18	wex 1812	. . . . 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 1568	. . 3 wff ∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
25	24, 1	wal 1568	. 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 1812	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 used by:  zfac  10462  ac2  10463