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Theorem zfreg 8042
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 8184). (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 8041 . 2
3 n0 3794 . 2
4 disj 3867 . . 3
54rexbii 2959 . 2
62, 3, 53imtr4i 266 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109  i^icin 3474   c0 3784
This theorem is referenced by:  en3lp  8054  inf3lem3  8068  setindtr  30966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-reg 8039
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785
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