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Theorem zfreg 8115
 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 8221). (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 8114 . 2
3 n0 3743 . 2
4 disj 3807 . . 3
54rexbii 2891 . 2
62, 3, 53imtr4i 270 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1446  wex 1665   wcel 1889   wne 2624  wral 2739  wrex 2740  cvv 3047   cin 3405  c0 3733 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-reg 8112 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-v 3049  df-dif 3409  df-in 3413  df-nul 3734 This theorem is referenced by:  en3lp  8126  inf3lem3  8140  setindtr  35891
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