MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nordeq Structured version   Visualization version   GIF version

Theorem nordeq 6403
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 6402 . . . 4 (Ord 𝐴 → ¬ 𝐴𝐴)
2 eleq1 2829 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴𝐵𝐴))
32notbid 318 . . . 4 (𝐴 = 𝐵 → (¬ 𝐴𝐴 ↔ ¬ 𝐵𝐴))
41, 3syl5ibcom 245 . . 3 (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵𝐴))
54necon2ad 2955 . 2 (Ord 𝐴 → (𝐵𝐴𝐴𝐵))
65imp 406 1 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  Ord word 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by:  php  9247  phplem1OLD  9254  phpOLD  9259  nogt01o  27741  ordtop  36437  limsucncmpi  36446
  Copyright terms: Public domain W3C validator