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Theorem onirri 5977
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 ∈ On
Assertion
Ref Expression
onirri ¬ 𝐴𝐴

Proof of Theorem onirri
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onordi 5975 . 2 Ord 𝐴
3 ordirr 5884 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2145  Ord word 5865  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  onssnel2i  5981  onuninsuci  7187  oelim2  7829  omopthlem2  7890  harndom  8625  wfelirr  8852  carduni  9007  pm54.43  9026  alephle  9111  alephfp  9131  pwxpndom2  9689  onsucsuccmpi  32779  onint1  32785  finxpreclem5  33569  wepwsolem  38138
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