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Theorem nordeq 5885
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 5884 . . . 4 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq1 2838 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
32notbid 307 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐵𝐴))
41, 3syl5ibcom 235 . . 3 (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
54necon2ad 2958 . 2 (Ord 𝐴 → (𝐵𝐴𝐴𝐵))
65imp 393 1 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  Ord word 5865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-eprel 5162  df-fr 5208  df-we 5210  df-ord 5869
This theorem is referenced by:  phplem1  8295  php  8300  ordtop  32772  limsucncmpi  32781
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