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Theorem onirri 5732
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 5730 . 2 Ord 𝐴
3 ordirr 5639 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3ax-mp 5 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 1975  Ord word 5620  Oncon0 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-tr 4670  df-eprel 4934  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-ord 5624  df-on 5625
This theorem is referenced by:  onssnel2i  5736  onuninsuci  6904  oelim2  7534  omopthlem2  7595  harndom  8324  wfelirr  8543  carduni  8662  pm54.43  8681  alephle  8766  alephfp  8786  pwxpndom2  9338  fvnobday  30882  onsucsuccmpi  31413  onint1  31419  finxpreclem5  32206  wepwsolem  36428
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