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

Theorem tz7.2 4377
 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 4122 . . 3
2 efrirr 4374 . . . . 5
3 eleq1 2343 . . . . . 6
43notbid 285 . . . . 5
52, 4syl5ibrcom 213 . . . 4
65necon2ad 2494 . . 3
71, 6anim12ii 553 . 2
873impia 1148 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684   wne 2446   wss 3152   wtr 4113   cep 4303   wfr 4349 This theorem is referenced by:  tz7.7  4418  trelpss  27660 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-fr 4352
 Copyright terms: Public domain W3C validator