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Theorem nordeq 4528
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 4526 . . . 4  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq1 2233 . . . . 5  |-  ( A  =  B  ->  ( A  e.  A  <->  B  e.  A ) )
32notbid 662 . . . 4  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  B  e.  A ) )
41, 3syl5ibcom 154 . . 3  |-  ( Ord 
A  ->  ( A  =  B  ->  -.  B  e.  A ) )
54necon2ad 2397 . 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 1348    e. wcel 2141    =/= wne 2340   Ord word 4347
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-sn 3589
This theorem is referenced by:  phplem1  6830
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