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Theorem nordeq 4504
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 4502 . . . 4  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq1 2220 . . . . 5  |-  ( A  =  B  ->  ( A  e.  A  <->  B  e.  A ) )
32notbid 657 . . . 4  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  B  e.  A ) )
41, 3syl5ibcom 154 . . 3  |-  ( Ord 
A  ->  ( A  =  B  ->  -.  B  e.  A ) )
54necon2ad 2384 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  A  =/= 
B ) )
65imp 123 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128    =/= wne 2327   Ord word 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-sn 3566
This theorem is referenced by:  phplem1  6798
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