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Axiom ax-ac2 8085
Description: In order to avoid uses of ax-reg 7302 for derivation of AC equivalents, we provide ax-ac2 8085, 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 8088. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1534 available. The derivation of ax-ac2 8085 from ax-ac 8081 is shown by theorem axac2 8089, and the reverse derivation by axac 8090. Note that we use ax-reg 7302 to derive ax-ac 8081 from ax-ac2 8085, but not to derive ax-ac2 8085 from ax-ac 8081. (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  set  y
2 vx . . . . . . . 8  set  x
31, 2wel 1686 . . . . . . 7  wff  y  e.  x
4 vz . . . . . . . . 9  set  z
54, 1wel 1686 . . . . . . . 8  wff  z  e.  y
6 vv . . . . . . . . . . 11  set  v
76, 2wel 1686 . . . . . . . . . 10  wff  v  e.  x
81, 6weq 1625 . . . . . . . . . . 11  wff  y  =  v
98wn 5 . . . . . . . . . 10  wff  -.  y  =  v
107, 9wa 360 . . . . . . . . 9  wff  ( v  e.  x  /\  -.  y  =  v )
114, 6wel 1686 . . . . . . . . 9  wff  z  e.  v
1210, 11wa 360 . . . . . . . 8  wff  ( ( v  e.  x  /\  -.  y  =  v
)  /\  z  e.  v )
135, 12wi 6 . . . . . . 7  wff  ( z  e.  y  ->  (
( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v ) )
143, 13wa 360 . . . . . 6  wff  ( y  e.  x  /\  (
z  e.  y  -> 
( ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v )
) )
153wn 5 . . . . . . 7  wff  -.  y  e.  x
164, 2wel 1686 . . . . . . . 8  wff  z  e.  x
176, 4wel 1686 . . . . . . . . . 10  wff  v  e.  z
186, 1wel 1686 . . . . . . . . . 10  wff  v  e.  y
1917, 18wa 360 . . . . . . . . 9  wff  ( v  e.  z  /\  v  e.  y )
20 vu . . . . . . . . . . . 12  set  u
2120, 4wel 1686 . . . . . . . . . . 11  wff  u  e.  z
2220, 1wel 1686 . . . . . . . . . . 11  wff  u  e.  y
2321, 22wa 360 . . . . . . . . . 10  wff  ( u  e.  z  /\  u  e.  y )
2420, 6weq 1625 . . . . . . . . . 10  wff  u  =  v
2523, 24wi 6 . . . . . . . . 9  wff  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v )
2619, 25wa 360 . . . . . . . 8  wff  ( ( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) )
2716, 26wi 6 . . . . . . 7  wff  ( z  e.  x  ->  (
( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) )
2815, 27wa 360 . . . . . 6  wff  ( -.  y  e.  x  /\  ( z  e.  x  ->  ( ( v  e.  z  /\  v  e.  y )  /\  (
( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )
2914, 28wo 359 . . . . 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 1528 . . . 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 1529 . . 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 1528 . 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 1529 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 set class
This axiom is referenced by:  axac3  8086
  Copyright terms: Public domain W3C validator