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Theorem onirri 4830
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  |-  A  e.  On
Assertion
Ref Expression
onirri  |-  -.  A  e.  A

Proof of Theorem onirri
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onordi 4828 . 2  |-  Ord  A
3 ordirr 4742 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
42, 3ax-mp 5 1  |-  -.  A  e.  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1756   Ord word 4723   Oncon0 4724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728
This theorem is referenced by:  onssnel2i  4834  onuninsuci  6456  oelim2  7039  omopthlem2  7100  harndom  7784  wfelirr  8037  carduni  8156  pm54.43  8175  alephle  8263  alephfp  8283  pwxpndom2  8837  fvnobday  27828  onsucsuccmpi  28294  onint1  28300  wepwsolem  29399
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