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Axiom ax-ac2 8299
Description: In order to avoid uses of ax-reg 7516 for derivation of AC equivalents, we provide ax-ac2 8299, 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 8301. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1552 available. The derivation of ax-ac2 8299 from ax-ac 8295 is shown by theorem axac2 8302, and the reverse derivation by axac 8303. Note that we use ax-reg 7516 to derive ax-ac 8295 from ax-ac2 8299, but not to derive ax-ac2 8299 from ax-ac 8295. (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 1722 . . . . . . 7  wff  y  e.  x
4 vz . . . . . . . . 9  set  z
54, 1wel 1722 . . . . . . . 8  wff  z  e.  y
6 vv . . . . . . . . . . 11  set  v
76, 2wel 1722 . . . . . . . . . 10  wff  v  e.  x
81, 6weq 1650 . . . . . . . . . . 11  wff  y  =  v
98wn 3 . . . . . . . . . 10  wff  -.  y  =  v
107, 9wa 359 . . . . . . . . 9  wff  ( v  e.  x  /\  -.  y  =  v )
114, 6wel 1722 . . . . . . . . 9  wff  z  e.  v
1210, 11wa 359 . . . . . . . 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 359 . . . . . 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 1722 . . . . . . . 8  wff  z  e.  x
176, 4wel 1722 . . . . . . . . . 10  wff  v  e.  z
186, 1wel 1722 . . . . . . . . . 10  wff  v  e.  y
1917, 18wa 359 . . . . . . . . 9  wff  ( v  e.  z  /\  v  e.  y )
20 vu . . . . . . . . . . . 12  set  u
2120, 4wel 1722 . . . . . . . . . . 11  wff  u  e.  z
2220, 1wel 1722 . . . . . . . . . . 11  wff  u  e.  y
2321, 22wa 359 . . . . . . . . . 10  wff  ( u  e.  z  /\  u  e.  y )
2420, 6weq 1650 . . . . . . . . . 10  wff  u  =  v
2523, 24wi 4 . . . . . . . . 9  wff  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v )
2619, 25wa 359 . . . . . . . 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 359 . . . . . 6  wff  ( -.  y  e.  x  /\  ( z  e.  x  ->  ( ( v  e.  z  /\  v  e.  y )  /\  (
( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) )
2914, 28wo 358 . . . . 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 1546 . . . 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 1547 . . 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 1546 . 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 1547 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  8300
  Copyright terms: Public domain W3C validator