Axiom ax-ac 10499

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The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥.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 10502 for a more detailed explanation. Theorem ac2 10501 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 10505 is slightly shorter when the biconditional of ax-ac 10499 is expanded into implication and negation. In axac3 10504 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 10721 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 10532, ac5 10517, and ac7 10513. The Axiom of Regularity ax-reg 9632 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as Theorem dfac2b 10171. Equivalents to AC are the well-ordering theorem weth 10535 and Zorn's lemma zorn 10547. See ac4 10515 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 9632 for derivation of AC equivalents, we provide ax-ac2 10503 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 10503 from ax-ac 10499 is shown by Theorem axac2 10506, and the reverse derivation by axac 10507. Therefore, new proofs should normally use ax-ac2 10503 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  10500  ac2  10501