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Theorem nordeq 4295
 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 4293 . . . 4 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq1 2142 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
32notbid 625 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐵𝐴))
41, 3syl5ibcom 153 . . 3 (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
54necon2ad 2303 . 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 1285   ∈ wcel 1434   ≠ wne 2246  Ord word 4125 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-sn 3412 This theorem is referenced by:  phplem1  6387
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