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

Theorem nordeq 4603
Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
nordeq  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )

Proof of Theorem nordeq
StepHypRef Expression
1 ordirr 4602 . . . 4  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq1 2498 . . . . 5  |-  ( A  =  B  ->  ( A  e.  A  <->  B  e.  A ) )
32notbid 287 . . . 4  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  B  e.  A ) )
41, 3syl5ibcom 213 . . 3  |-  ( Ord 
A  ->  ( A  =  B  ->  -.  B  e.  A ) )
54necon2ad 2654 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  A  =/= 
B ) )
65imp 420 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   Ord word 4583
This theorem is referenced by:  phplem1  7289  php  7294  ordtop  26191  limsucncmpi  26200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-fr 4544  df-we 4546  df-ord 4587
  Copyright terms: Public domain W3C validator