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Theorem onirri 6478
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 6476 . 2 Ord 𝐴
3 ordirr 6383 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  Ord word 6364  Oncon0 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  onssnel2i  6482  onuninsuci  7829  nlim2  8490  ord1eln01  8496  ord2eln012  8497  oelim2  8595  omopthlem2  8659  enpr2dOLD  9050  harndom  9557  ssttrcl  9710  wfelirr  9820  carduni  9976  pm54.43  9996  alephle  10083  alephfp  10103  pwxpndom2  10660  oldirr  27384  lrrecpo  27425  onsucsuccmpi  35328  onint1  35334  finxpreclem5  36276  wepwsolem  41784
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