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Axiom ax-ac2 8864
Description: In order to avoid uses of ax-reg 8039 for derivation of AC equivalents, we provide ax-ac2 8864, 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 8866. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1618 available. The derivation of ax-ac2 8864 from ax-ac 8860 is shown by theorem axac2 8867, and the reverse derivation by axac 8868. Note that we use ax-reg 8039 to derive ax-ac 8860 from ax-ac2 8864, but not to derive ax-ac2 8864 from ax-ac 8860. (Contributed by NM, 19-Dec-2016.)
Assertion
Ref Expression
ax-ac2
Distinct variable group:   , , , ,

Detailed syntax breakdown of Axiom ax-ac2
StepHypRef Expression
1 vy . . . . . . . 8
2 vx . . . . . . . 8
31, 2wel 1819 . . . . . . 7
4 vz . . . . . . . . 9
54, 1wel 1819 . . . . . . . 8
6 vv . . . . . . . . . . 11
76, 2wel 1819 . . . . . . . . . 10
81, 6weq 1733 . . . . . . . . . . 11
98wn 3 . . . . . . . . . 10
107, 9wa 369 . . . . . . . . 9
114, 6wel 1819 . . . . . . . . 9
1210, 11wa 369 . . . . . . . 8
135, 12wi 4 . . . . . . 7
143, 13wa 369 . . . . . 6
153wn 3 . . . . . . 7
164, 2wel 1819 . . . . . . . 8
176, 4wel 1819 . . . . . . . . . 10
186, 1wel 1819 . . . . . . . . . 10
1917, 18wa 369 . . . . . . . . 9
20 vu . . . . . . . . . . . 12
2120, 4wel 1819 . . . . . . . . . . 11
2220, 1wel 1819 . . . . . . . . . . 11
2321, 22wa 369 . . . . . . . . . 10
2420, 6weq 1733 . . . . . . . . . 10
2523, 24wi 4 . . . . . . . . 9
2619, 25wa 369 . . . . . . . 8
2716, 26wi 4 . . . . . . 7
2815, 27wa 369 . . . . . 6
2914, 28wo 368 . . . . 5
3029, 20wal 1393 . . . 4
3130, 6wex 1612 . . 3
3231, 4wal 1393 . 2
3332, 1wex 1612 1
Colors of variables: wff setvar class
This axiom is referenced by:  axac3  8865
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