Axiom ax-ac2 8877

Description: In order to avoid uses of ax-reg 8054 for derivation of AC equivalents, we provide ax-ac2 8877, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 8879. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1641 available. The derivation of ax-ac2 8877 from ax-ac 8873 is shown by theorem axac2 8880, and the reverse derivation by axac 8881. Note that we use ax-reg 8054 to derive ax-ac 8873 from ax-ac2 8877, but not to derive ax-ac2 8877 from ax-ac 8873. (Contributed by NM, 19-Dec-2016.)

Assertion

Ref	Expression
ax-ac2	|- E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )

Distinct variable group:   x, y, z, v, u

Detailed syntax breakdown of Axiom ax-ac2

Step	Hyp	Ref	Expression
1		vy	. . . . . . . 8 setvar y
2		vx	. . . . . . . 8 setvar x
3	1, 2	wel 1845	. . . . . . 7 wff y e. x
4		vz	. . . . . . . . 9 setvar z
5	4, 1	wel 1845	. . . . . . . 8 wff z e. y
6		vv	. . . . . . . . . . 11 setvar v
7	6, 2	wel 1845	. . . . . . . . . 10 wff v e. x
8	1, 6	weq 1759	. . . . . . . . . . 11 wff y = v
9	8	wn 3	. . . . . . . . . 10 wff -. y = v
10	7, 9	wa 369	. . . . . . . . 9 wff ( v e. x /\ -. y = v )
11	4, 6	wel 1845	. . . . . . . . 9 wff z e. v
12	10, 11	wa 369	. . . . . . . 8 wff ( ( v e. x /\ -. y = v ) /\ z e. v )
13	5, 12	wi 4	. . . . . . 7 wff ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) )
14	3, 13	wa 369	. . . . . 6 wff ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) )
15	3	wn 3	. . . . . . 7 wff -. y e. x
16	4, 2	wel 1845	. . . . . . . 8 wff z e. x
17	6, 4	wel 1845	. . . . . . . . . 10 wff v e. z
18	6, 1	wel 1845	. . . . . . . . . 10 wff v e. y
19	17, 18	wa 369	. . . . . . . . 9 wff ( v e. z /\ v e. y )
20		vu	. . . . . . . . . . . 12 setvar u
21	20, 4	wel 1845	. . . . . . . . . . 11 wff u e. z
22	20, 1	wel 1845	. . . . . . . . . . 11 wff u e. y
23	21, 22	wa 369	. . . . . . . . . 10 wff ( u e. z /\ u e. y )
24	20, 6	weq 1759	. . . . . . . . . 10 wff u = v
25	23, 24	wi 4	. . . . . . . . 9 wff ( ( u e. z /\ u e. y ) -> u = v )
26	19, 25	wa 369	. . . . . . . 8 wff ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) )
27	16, 26	wi 4	. . . . . . 7 wff ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) )
28	15, 27	wa 369	. . . . . 6 wff ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) )
29	14, 28	wo 368	. . . . 5 wff ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
30	29, 20	wal 1405	. . . 4 wff A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
31	30, 6	wex 1635	. . 3 wff E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
32	31, 4	wal 1405	. 2 wff A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
33	32, 1	wex 1635	1 wff E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )

This axiom is referenced by:  axac3  8878