Axiom ax-ac2 8843

Description: In order to avoid uses of ax-reg 8018 for derivation of AC equivalents, we provide ax-ac2 8843, 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 8845. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1601 available. The derivation of ax-ac2 8843 from ax-ac 8839 is shown by theorem axac2 8846, and the reverse derivation by axac 8847. Note that we use ax-reg 8018 to derive ax-ac 8839 from ax-ac2 8843, but not to derive ax-ac2 8843 from ax-ac 8839. (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 1768	. . . . . . 7 wff y e. x
4		vz	. . . . . . . . 9 setvar z
5	4, 1	wel 1768	. . . . . . . 8 wff z e. y
6		vv	. . . . . . . . . . 11 setvar v
7	6, 2	wel 1768	. . . . . . . . . 10 wff v e. x
8	1, 6	weq 1705	. . . . . . . . . . 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 1768	. . . . . . . . 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 1768	. . . . . . . 8 wff z e. x
17	6, 4	wel 1768	. . . . . . . . . 10 wff v e. z
18	6, 1	wel 1768	. . . . . . . . . 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 1768	. . . . . . . . . . 11 wff u e. z
22	20, 1	wel 1768	. . . . . . . . . . 11 wff u e. y
23	21, 22	wa 369	. . . . . . . . . 10 wff ( u e. z /\ u e. y )
24	20, 6	weq 1705	. . . . . . . . . 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 1377	. . . 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 1596	. . 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 1377	. 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 1596	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  8844