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Theorem zfreg 7599
 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 be a set, that can be proved with more difficulty (see zfregs 7704). (Contributed by NM, 26-Nov-1995.)
Hypothesis
Ref Expression
zfreg.1
Assertion
Ref Expression
zfreg
Distinct variable group:   ,

Proof of Theorem zfreg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 zfreg.1 . . 3
21zfregcl 7598 . 2
3 n0 3625 . 2
4 disj 3696 . . 3
54rexbii 2737 . 2
62, 3, 53imtr4i 259 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4  wex 1551   wceq 1654   wcel 1728   wne 2606  wral 2712  wrex 2713  cvv 2965   cin 3308  c0 3616 This theorem is referenced by:  inf3lem3  7621  en3lp  7708  setindtr  27207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-reg 7596 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-v 2967  df-dif 3312  df-in 3316  df-nul 3617
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