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Theorem nordeq 4360
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 4358 . . . 4  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq1 2150 . . . . 5  |-  ( A  =  B  ->  ( A  e.  A  <->  B  e.  A ) )
32notbid 627 . . . 4  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  B  e.  A ) )
41, 3syl5ibcom 153 . . 3  |-  ( Ord 
A  ->  ( A  =  B  ->  -.  B  e.  A ) )
54necon2ad 2312 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  A  =/= 
B ) )
65imp 122 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    =/= wne 2255   Ord word 4189
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3452
This theorem is referenced by:  phplem1  6568
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