Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfreg2 Unicode version

Theorem zfreg2 7326
 Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 7325) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480. (Contributed by NM, 17-Sep-2003.)
Hypothesis
Ref Expression
zfreg2.1
Assertion
Ref Expression
zfreg2
Distinct variable group:   ,

Proof of Theorem zfreg2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 zfreg2.1 . . 3
21zfregcl 7324 . 2
3 n0 3477 . 2
4 disjr 3509 . . 3
54rexbii 2581 . 2
62, 3, 53imtr4i 257 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4  wex 1531   wceq 1632   wcel 1696   wne 2459  wral 2556  wrex 2557  cvv 2801   cin 3164  c0 3468 This theorem is referenced by:  zfregfr  7332 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-reg 7322 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-nul 3469
 Copyright terms: Public domain W3C validator