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Theorem zfreg 7193
 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 7298). (Contributed by NM, 26-Nov-1995.)
Hypothesis
Ref Expression
zfreg.1
Assertion
Ref Expression
zfreg
Distinct variable group:   ,

Proof of Theorem zfreg
StepHypRef Expression
1 zfreg.1 . . 3
21zfregcl 7192 . 2
3 n0 3371 . 2
4 disj 3402 . . 3
54rexbii 2532 . 2
62, 3, 53imtr4i 259 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6  wex 1537   wceq 1619   wcel 1621   wne 2412  wral 2509  wrex 2510  cvv 2727   cin 3077  c0 3362 This theorem is referenced by:  inf3lem3  7215  en3lp  7302  setindtr  26283 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 1926  ax-ext 2234  ax-reg 7190 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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-in 3085  df-nul 3363
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