Axiom ax-ac2 8846

Description: In order to avoid uses of ax-reg 8021 for derivation of AC equivalents, we provide ax-ac2 8846, 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 8848. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1605 available. The derivation of ax-ac2 8846 from ax-ac 8842 is shown by theorem axac2 8849, and the reverse derivation by axac 8850. Note that we use ax-reg 8021 to derive ax-ac 8842 from ax-ac2 8846, but not to derive ax-ac2 8846 from ax-ac 8842. (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 1805	. . . . . . 7 wff y e. x
4		vz	. . . . . . . . 9 setvar z
5	4, 1	wel 1805	. . . . . . . 8 wff z e. y
6		vv	. . . . . . . . . . 11 setvar v
7	6, 2	wel 1805	. . . . . . . . . 10 wff v e. x
8	1, 6	weq 1720	. . . . . . . . . . 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 1805	. . . . . . . . 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 1805	. . . . . . . 8 wff z e. x
17	6, 4	wel 1805	. . . . . . . . . 10 wff v e. z
18	6, 1	wel 1805	. . . . . . . . . 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 1805	. . . . . . . . . . 11 wff u e. z
22	20, 1	wel 1805	. . . . . . . . . . 11 wff u e. y
23	21, 22	wa 369	. . . . . . . . . 10 wff ( u e. z /\ u e. y )
24	20, 6	weq 1720	. . . . . . . . . 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 1381	. . . 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 1599	. . 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 1381	. 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 1599	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  8847