Axiom ax-ac 4890

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The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each non-empty member of x. One entry in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 4893 for a more detailed explanation.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 4928 is slightly shorter when the biconditional of ax-ac 4890 is expanded into implication and negation.

Standard textbook versions of AC are derived as ac8 4909, ac5 4898, and ac7 4894. The Axiom of Regularity ax-reg 4736 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 4888. Equivalents to AC are the well-ordering theorem weth 4933 and Zorn's lemma zorn 4943. See ac4 4896 for comments about stronger versions of AC.

Assertion

Ref	Expression
ax-ac	|- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. 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 991	. . . . . . 7 class z
3		vw	. . . . . . . 8 set w
4	3	cv 991	. . . . . . 7 class w
5	2, 4	wcel 994	. . . . . 6 wff z e. w
6		vx	. . . . . . . 8 set x
7	6	cv 991	. . . . . . 7 class x
8	4, 7	wcel 994	. . . . . 6 wff w e. x
9	5, 8	wa 221	. . . . 5 wff (z e. w /\ w e. x)
10		vu	. . . . . . . . . . . . 13 set u
11	10	cv 991	. . . . . . . . . . . 12 class u
12	11, 4	wcel 994	. . . . . . . . . . 11 wff u e. w
13		vt	. . . . . . . . . . . . 13 set t
14	13	cv 991	. . . . . . . . . . . 12 class t
15	4, 14	wcel 994	. . . . . . . . . . 11 wff w e. t
16	12, 15	wa 221	. . . . . . . . . 10 wff (u e. w /\ w e. t)
17	11, 14	wcel 994	. . . . . . . . . . 11 wff u e. t
18		vy	. . . . . . . . . . . . 13 set y
19	18	cv 991	. . . . . . . . . . . 12 class y
20	14, 19	wcel 994	. . . . . . . . . . 11 wff t e. y
21	17, 20	wa 221	. . . . . . . . . 10 wff (u e. t /\ t e. y)
22	16, 21	wa 221	. . . . . . . . 9 wff ((u e. w /\ w e. t) /\ (u e. t /\ t e. y))
23	22, 13	wex 1016	. . . . . . . 8 wff E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y))
24		vv	. . . . . . . . . 10 set v
25	24	cv 991	. . . . . . . . 9 class v
26	11, 25	wceq 992	. . . . . . . 8 wff u = v
27	23, 26	wb 144	. . . . . . 7 wff (E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)
28	27, 10	wal 990	. . . . . 6 wff A.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)
29	28, 24	wex 1016	. . . . 5 wff E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)
30	9, 29	wi 3	. . . 4 wff ((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
31	30, 3	wal 990	. . 3 wff A.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
32	31, 1	wal 990	. 2 wff A.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
33	32, 18	wex 1016	1 wff E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))

This axiom is referenced by:  zfac 4891  ac2 4892

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