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Theorem nordeq 4493
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 4492 . . . 4  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq1 2418 . . . . 5  |-  ( A  =  B  ->  ( A  e.  A  <->  B  e.  A ) )
32notbid 285 . . . 4  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  B  e.  A ) )
41, 3syl5ibcom 211 . . 3  |-  ( Ord 
A  ->  ( A  =  B  ->  -.  B  e.  A ) )
54necon2ad 2569 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  A  =/= 
B ) )
65imp 418 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   Ord word 4473
This theorem is referenced by:  phplem1  7128  php  7133  ordtop  25434  limsucncmpi  25443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-eprel 4387  df-fr 4434  df-we 4436  df-ord 4477
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