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Theorem onirri 4372
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
onirri.1 𝐴 ∈ On
Assertion
Ref Expression
onirri ¬ 𝐴𝐴

Proof of Theorem onirri
StepHypRef Expression
1 onirri.1 . . 3 𝐴 ∈ On
21onordi 4262 . 2 Ord 𝐴
3 ordirr 4371 . 2 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3ax-mp 7 1 ¬ 𝐴𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1439  Ord word 4198  Oncon0 4199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-sn 3456  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204
This theorem is referenced by:  pm54.43  6879
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