Axiom ax-ac2 8911

Description: In order to avoid uses of ax-reg 8125 for derivation of AC equivalents, we provide ax-ac2 8911, 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 8913. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1677 available. The derivation of ax-ac2 8911 from ax-ac 8907 is shown by theorem axac2 8914, and the reverse derivation by axac 8915. Note that we use ax-reg 8125 to derive ax-ac 8907 from ax-ac2 8911, but not to derive ax-ac2 8911 from ax-ac 8907. (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 1905	. . . . . . 7 wff y e. x
4		vz	. . . . . . . . 9 setvar z
5	4, 1	wel 1905	. . . . . . . 8 wff z e. y
6		vv	. . . . . . . . . . 11 setvar v
7	6, 2	wel 1905	. . . . . . . . . 10 wff v e. x
8	1, 6	weq 1799	. . . . . . . . . . 11 wff y = v
9	8	wn 3	. . . . . . . . . 10 wff -. y = v
10	7, 9	wa 376	. . . . . . . . 9 wff ( v e. x /\ -. y = v )
11	4, 6	wel 1905	. . . . . . . . 9 wff z e. v
12	10, 11	wa 376	. . . . . . . 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 376	. . . . . 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 1905	. . . . . . . 8 wff z e. x
17	6, 4	wel 1905	. . . . . . . . . 10 wff v e. z
18	6, 1	wel 1905	. . . . . . . . . 10 wff v e. y
19	17, 18	wa 376	. . . . . . . . 9 wff ( v e. z /\ v e. y )
20		vu	. . . . . . . . . . . 12 setvar u
21	20, 4	wel 1905	. . . . . . . . . . 11 wff u e. z
22	20, 1	wel 1905	. . . . . . . . . . 11 wff u e. y
23	21, 22	wa 376	. . . . . . . . . 10 wff ( u e. z /\ u e. y )
24	20, 6	weq 1799	. . . . . . . . . 10 wff u = v
25	23, 24	wi 4	. . . . . . . . 9 wff ( ( u e. z /\ u e. y ) -> u = v )
26	19, 25	wa 376	. . . . . . . 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 376	. . . . . 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 375	. . . . 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 1450	. . . 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 1671	. . 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 1450	. 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 1671	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  8912