Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.2 Unicode version

Theorem tz7.2 4868
 Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent . (Contributed by NM, 4-May-1994.)
Assertion
Ref Expression
tz7.2

Proof of Theorem tz7.2
StepHypRef Expression
1 trss 4554 . . 3
2 efrirr 4865 . . . . 5
3 eleq1 2529 . . . . . 6
43notbid 294 . . . . 5
52, 4syl5ibrcom 222 . . . 4
65necon2ad 2670 . . 3
71, 6anim12ii 570 . 2
873impia 1193 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  C_wss 3475  Trwtr 4545   cep 4794  Frwfr 4840 This theorem is referenced by:  tz7.7  4909  trelpss  31364 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-fr 4843
 Copyright terms: Public domain W3C validator