Axiom ax-ac 4716

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The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each non-empty member of x. One entry in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 4719 for a more detailed explanation.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 4754 is slightly shorter when the biconditional of ax-ac 4716 is expanded into implication and negation.

Standard textbook versions of AC are derived as ac8 4735, ac5 4724, and ac7 4720. The Axiom of Regularity ax-reg 4565 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 4714. Equivalents to AC are the well-ordering theorem weth 4759 and Zorn's lemma zorn 4769. See ac4 4722 for comments about stronger versions of AC.

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	. . . . . . . 8 set z
2	1	cv 952	. . . . . . 7 class z
3		vw	. . . . . . . 8 set w
4	3	cv 952	. . . . . . 7 class w
5	2, 4	wcel 955	. . . . . 6 wff z ∈ w
6		vx	. . . . . . . 8 set x
7	6	cv 952	. . . . . . 7 class x
8	4, 7	wcel 955	. . . . . 6 wff w ∈ x
9	5, 8	wa 223	. . . . 5 wff (z ∈ w ⋀ w ∈ x)
10		vu	. . . . . . . . . . . . 13 set u
11	10	cv 952	. . . . . . . . . . . 12 class u
12	11, 4	wcel 955	. . . . . . . . . . 11 wff u ∈ w
13		vt	. . . . . . . . . . . . 13 set t
14	13	cv 952	. . . . . . . . . . . 12 class t
15	4, 14	wcel 955	. . . . . . . . . . 11 wff w ∈ t
16	12, 15	wa 223	. . . . . . . . . 10 wff (u ∈ w ⋀ w ∈ t)
17	11, 14	wcel 955	. . . . . . . . . . 11 wff u ∈ t
18		vy	. . . . . . . . . . . . 13 set y
19	18	cv 952	. . . . . . . . . . . 12 class y
20	14, 19	wcel 955	. . . . . . . . . . 11 wff t ∈ y
21	17, 20	wa 223	. . . . . . . . . 10 wff (u ∈ t ⋀ t ∈ y)
22	16, 21	wa 223	. . . . . . . . 9 wff ((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y))
23	22, 13	wex 977	. . . . . . . 8 wff ∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y))
24		vv	. . . . . . . . . 10 set v
25	24	cv 952	. . . . . . . . 9 class v
26	11, 25	wceq 953	. . . . . . . 8 wff u = v
27	23, 26	wb 146	. . . . . . 7 wff (∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v)
28	27, 10	wal 951	. . . . . 6 wff ∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v)
29	28, 24	wex 977	. . . . 5 wff ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v)
30	9, 29	wi 3	. . . 4 wff ((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v))
31	30, 3	wal 951	. . 3 wff ∀w((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v))
32	31, 1	wal 951	. 2 wff ∀z∀w((z ∈ w ⋀ w ∈ x) → ∃v∀u(∃t((u ∈ w ⋀ w ∈ t) ⋀ (u ∈ t ⋀ t ∈ y)) ↔ u = v))
33	32, 18	wex 977	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:  axac 4717  ac2 4718

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