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Theorem nordeq 4561
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 4559 . . . 4 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq1 2252 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
32notbid 668 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐵𝐴))
41, 3syl5ibcom 155 . . 3 (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
54necon2ad 2417 . 2 (Ord 𝐴 → (𝐵𝐴𝐴𝐵))
65imp 124 1 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wcel 2160  wne 2360  Ord word 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-sn 3613
This theorem is referenced by:  phplem1  6881
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