Axiom ax-ac2 8895

Description: In order to avoid uses of ax-reg 8111 for derivation of AC equivalents, we provide ax-ac2 8895, 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 8897. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1666 available. The derivation of ax-ac2 8895 from ax-ac 8891 is shown by theorem axac2 8898, and the reverse derivation by axac 8899. Note that we use ax-reg 8111 to derive ax-ac 8891 from ax-ac2 8895, but not to derive ax-ac2 8895 from ax-ac 8891. (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 1870	. . . . . . 7 wff y e. x
4		vz	. . . . . . . . 9 setvar z
5	4, 1	wel 1870	. . . . . . . 8 wff z e. y
6		vv	. . . . . . . . . . 11 setvar v
7	6, 2	wel 1870	. . . . . . . . . 10 wff v e. x
8	1, 6	weq 1781	. . . . . . . . . . 11 wff y = v
9	8	wn 3	. . . . . . . . . 10 wff -. y = v
10	7, 9	wa 371	. . . . . . . . 9 wff ( v e. x /\ -. y = v )
11	4, 6	wel 1870	. . . . . . . . 9 wff z e. v
12	10, 11	wa 371	. . . . . . . 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 371	. . . . . 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 1870	. . . . . . . 8 wff z e. x
17	6, 4	wel 1870	. . . . . . . . . 10 wff v e. z
18	6, 1	wel 1870	. . . . . . . . . 10 wff v e. y
19	17, 18	wa 371	. . . . . . . . 9 wff ( v e. z /\ v e. y )
20		vu	. . . . . . . . . . . 12 setvar u
21	20, 4	wel 1870	. . . . . . . . . . 11 wff u e. z
22	20, 1	wel 1870	. . . . . . . . . . 11 wff u e. y
23	21, 22	wa 371	. . . . . . . . . 10 wff ( u e. z /\ u e. y )
24	20, 6	weq 1781	. . . . . . . . . 10 wff u = v
25	23, 24	wi 4	. . . . . . . . 9 wff ( ( u e. z /\ u e. y ) -> u = v )
26	19, 25	wa 371	. . . . . . . 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 371	. . . . . 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 370	. . . . 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 1436	. . . 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 1660	. . 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 1436	. 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 1660	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  8896