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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
StepHypRef Expression
1 vy . . . . . . . 8  setvar  y
2 vx . . . . . . . 8  setvar  x
31, 2wel 1805 . . . . . . 7  wff  y  e.  x
4 vz . . . . . . . . 9  setvar  z
54, 1wel 1805 . . . . . . . 8  wff  z  e.  y
6 vv . . . . . . . . . . 11  setvar  v
76, 2wel 1805 . . . . . . . . . 10  wff  v  e.  x
81, 6weq 1720 . . . . . . . . . . 11  wff  y  =  v
98wn 3 . . . . . . . . . 10  wff  -.  y  =  v
107, 9wa 369 . . . . . . . . 9  wff  ( v  e.  x  /\  -.  y  =  v )
114, 6wel 1805 . . . . . . . . 9  wff  z  e.  v
1210, 11wa 369 . . . . . . . 8  wff  ( ( v  e.  x  /\  -.  y  =  v
)  /\  z  e.  v )
135, 12wi 4 . . . . . . 7  wff  ( z  e.  y  ->  (
( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v ) )
143, 13wa 369 . . . . . 6  wff  ( y  e.  x  /\  (
z  e.  y  -> 
( ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v )
) )
153wn 3 . . . . . . 7  wff  -.  y  e.  x
164, 2wel 1805 . . . . . . . 8  wff  z  e.  x
176, 4wel 1805 . . . . . . . . . 10  wff  v  e.  z
186, 1wel 1805 . . . . . . . . . 10  wff  v  e.  y
1917, 18wa 369 . . . . . . . . 9  wff  ( v  e.  z  /\  v  e.  y )
20 vu . . . . . . . . . . . 12  setvar  u
2120, 4wel 1805 . . . . . . . . . . 11  wff  u  e.  z
2220, 1wel 1805 . . . . . . . . . . 11  wff  u  e.  y
2321, 22wa 369 . . . . . . . . . 10  wff  ( u  e.  z  /\  u  e.  y )
2420, 6weq 1720 . . . . . . . . . 10  wff  u  =  v
2523, 24wi 4 . . . . . . . . 9  wff  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v )
2619, 25wa 369 . . . . . . . 8  wff  ( ( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) )
2716, 26wi 4 . . . . . . 7  wff  ( z  e.  x  ->  (
( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) )
2815, 27wa 369 . . . . . 6  wff  ( -.  y  e.  x  /\  ( z  e.  x  ->  ( ( v  e.  z  /\  v  e.  y )  /\  (
( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )
2914, 28wo 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 ) ) ) ) )
3029, 20wal 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 ) ) ) ) )
3130, 6wex 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 ) ) ) ) )
3231, 4wal 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 ) ) ) ) )
3332, 1wex 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 ) ) ) ) )
Colors of variables: wff setvar class
This axiom is referenced by:  axac3  8847
  Copyright terms: Public domain W3C validator