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Theorem onirri 5751
 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 5749 . 2 Ord 𝐴
3 ordirr 5658 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3ax-mp 5 1 ¬ 𝐴𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1977  Ord word 5639  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  onssnel2i  5755  onuninsuci  6932  oelim2  7562  omopthlem2  7623  harndom  8352  wfelirr  8571  carduni  8690  pm54.43  8709  alephle  8794  alephfp  8814  pwxpndom2  9366  fvnobday  31081  onsucsuccmpi  31612  onint1  31618  finxpreclem5  32408  wepwsolem  36630
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