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

Theorem onirri 4990
 Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1
Assertion
Ref Expression
onirri

Proof of Theorem onirri
StepHypRef Expression
1 on.1 . . 3
21onordi 4988 . 2
3 ordirr 4902 . 2
42, 3ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wcel 1767   word 4883  con0 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888 This theorem is referenced by:  onssnel2i  4994  onuninsuci  6670  oelim2  7256  omopthlem2  7317  harndom  8002  wfelirr  8255  carduni  8374  pm54.43  8393  alephle  8481  alephfp  8501  pwxpndom2  9055  fvnobday  29369  onsucsuccmpi  29835  onint1  29841  wepwsolem  30915
 Copyright terms: Public domain W3C validator