Axiom ax-ac 8856

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The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each nonempty member of x. One entry in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 8859 for a more detailed explanation. Theorem ac2 8858 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 8862 is slightly shorter when the biconditional of ax-ac 8856 is expanded into implication and negation. In axac3 8861 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9076 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8889, ac5 8874, and ac7 8870. The Axiom of Regularity ax-reg 8036 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8528. Equivalents to AC are the well-ordering theorem weth 8892 and Zorn's lemma zorn 8904. See ac4 8872 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8036 for derivation of AC equivalents, we provide ax-ac2 8860 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8860 from ax-ac 8856 is shown by theorem axac2 8863, and the reverse derivation by axac 8864. Therefore, new proofs should normally use ax-ac2 8860 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Assertion

Ref	Expression
ax-ac	|- E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. 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	. . . . . . 7 setvar z
2		vw	. . . . . . 7 setvar w
3	1, 2	wel 1820	. . . . . 6 wff z e. w
4		vx	. . . . . . 7 setvar x
5	2, 4	wel 1820	. . . . . 6 wff w e. x
6	3, 5	wa 369	. . . . 5 wff ( z e. w /\ w e. x )
7		vu	. . . . . . . . . . . 12 setvar u
8	7, 2	wel 1820	. . . . . . . . . . 11 wff u e. w
9		vt	. . . . . . . . . . . 12 setvar t
10	2, 9	wel 1820	. . . . . . . . . . 11 wff w e. t
11	8, 10	wa 369	. . . . . . . . . 10 wff ( u e. w /\ w e. t )
12	7, 9	wel 1820	. . . . . . . . . . 11 wff u e. t
13		vy	. . . . . . . . . . . 12 setvar y
14	9, 13	wel 1820	. . . . . . . . . . 11 wff t e. y
15	12, 14	wa 369	. . . . . . . . . 10 wff ( u e. t /\ t e. y )
16	11, 15	wa 369	. . . . . . . . 9 wff ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) )
17	16, 9	wex 1613	. . . . . . . 8 wff E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) )
18		vv	. . . . . . . . 9 setvar v
19	7, 18	weq 1734	. . . . . . . 8 wff u = v
20	17, 19	wb 184	. . . . . . 7 wff ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v )
21	20, 7	wal 1393	. . . . . 6 wff A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v )
22	21, 18	wex 1613	. . . . 5 wff E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v )
23	6, 22	wi 4	. . . 4 wff ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
24	23, 2	wal 1393	. . 3 wff A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
25	24, 1	wal 1393	. 2 wff A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
26	25, 13	wex 1613	1 wff E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )

This axiom is referenced by:  zfac  8857  ac2  8858