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Theorem zfreg 7277
Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that  A be a set, that can be proved with more difficulty (see zfregs 7382). (Contributed by NM, 26-Nov-1995.)
Hypothesis
Ref Expression
zfreg.1  |-  A  e. 
_V
Assertion
Ref Expression
zfreg  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Distinct variable group:    x, A

Proof of Theorem zfreg
StepHypRef Expression
1 zfreg.1 . . 3  |-  A  e. 
_V
21zfregcl 7276 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3439 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disj 3470 . . 3  |-  ( ( x  i^i  A )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2543 . 2  |-  ( E. x  e.  A  ( x  i^i  A )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 259 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763    i^i cin 3126   (/)c0 3430
This theorem is referenced by:  inf3lem3  7299  en3lp  7386  setindtr  26485
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-reg 7274
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-in 3134  df-nul 3431
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