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

Theorem zfreg2 8120
 Description: The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 8119) 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 8118 . 2
3 n0 3771 . 2
4 disjr 3836 . . 3
54rexbii 2924 . 2
62, 3, 53imtr4i 269 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1437  wex 1657   wcel 1872   wne 2614  wral 2771  wrex 2772  cvv 3080   cin 3435  c0 3761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-reg 8116 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-in 3443  df-nul 3762 This theorem is referenced by:  zfregfr  8126
 Copyright terms: Public domain W3C validator