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Theorem nordeq 4360
Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
nordeq ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)

Proof of Theorem nordeq
StepHypRef Expression
1 ordirr 4358 . . . 4 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq1 2150 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
32notbid 627 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐵𝐴))
41, 3syl5ibcom 153 . . 3 (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
54necon2ad 2312 . 2 (Ord 𝐴 → (𝐵𝐴𝐴𝐵))
65imp 122 1 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wcel 1438  wne 2255  Ord word 4189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3452
This theorem is referenced by:  phplem1  6566
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