Axiom ax-ac 5702

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The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each non-empty member of x. One entry in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 5705 for a more detailed explanation.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 5740 is slightly shorter when the biconditional of ax-ac 5702 is expanded into implication and negation.

Standard textbook versions of AC are derived as ac8 5721, ac5 5710, and ac7 5706. The Axiom of Regularity ax-reg 5505 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem aceq6b 5700. Equivalents to AC are the well-ordering theorem weth 5745 and Zorn's lemma zorn 5755. See ac4 5708 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 1135	. . . . . . 7 class z
3		vw	. . . . . . . 8 set w
4	3	cv 1135	. . . . . . 7 class w
5	2, 4	wcel 1138	. . . . . 6 wff z e. w
6		vx	. . . . . . . 8 set x
7	6	cv 1135	. . . . . . 7 class x
8	4, 7	wcel 1138	. . . . . 6 wff w e. x
9	5, 8	wa 239	. . . . 5 wff (z e. w /\ w e. x)
10		vu	. . . . . . . . . . . . 13 set u
11	10	cv 1135	. . . . . . . . . . . 12 class u
12	11, 4	wcel 1138	. . . . . . . . . . 11 wff u e. w
13		vt	. . . . . . . . . . . . 13 set t
14	13	cv 1135	. . . . . . . . . . . 12 class t
15	4, 14	wcel 1138	. . . . . . . . . . 11 wff w e. t
16	12, 15	wa 239	. . . . . . . . . 10 wff (u e. w /\ w e. t)
17	11, 14	wcel 1138	. . . . . . . . . . 11 wff u e. t
18		vy	. . . . . . . . . . . . 13 set y
19	18	cv 1135	. . . . . . . . . . . 12 class y
20	14, 19	wcel 1138	. . . . . . . . . . 11 wff t e. y
21	17, 20	wa 239	. . . . . . . . . 10 wff (u e. t /\ t e. y)
22	16, 21	wa 239	. . . . . . . . 9 wff ((u e. w /\ w e. t) /\ (u e. t /\ t e. y))
23	22, 13	wex 1164	. . . . . . . 8 wff E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y))
24		vv	. . . . . . . . . 10 set v
25	24	cv 1135	. . . . . . . . 9 class v
26	11, 25	wceq 1136	. . . . . . . 8 wff u = v
27	23, 26	wb 162	. . . . . . 7 wff (E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)
28	27, 10	wal 1134	. . . . . 6 wff A.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v)
29	28, 24	wex 1164	. . . . 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 1134	. . 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 1134	. 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 1164	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 5703  ac2 5704

Copyright terms: Public domain