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Theorem zfreg 7311
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 7416). (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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 zfreg.1 . . 3  |-  A  e. 
_V
21zfregcl 7310 . 2  |-  ( E. x  x  e.  A  ->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
3 n0 3466 . 2  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
4 disj 3497 . . 3  |-  ( ( x  i^i  A )  =  (/)  <->  A. y  e.  x  -.  y  e.  A
)
54rexbii 2570 . 2  |-  ( E. x  e.  A  ( x  i^i  A )  =  (/)  <->  E. x  e.  A  A. y  e.  x  -.  y  e.  A
)
62, 3, 53imtr4i 257 1  |-  ( A  =/=  (/)  ->  E. x  e.  A  ( x  i^i  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790    i^i cin 3153   (/)c0 3457
This theorem is referenced by:  inf3lem3  7333  en3lp  7420  setindtr  27128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-reg 7308
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-in 3161  df-nul 3458
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