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Theorem reapval 8531
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8543 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
reapval  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )

Proof of Theorem reapval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4008 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <  y  <->  A  <  B ) )
2 simpr 110 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
3 simpl 109 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
42, 3breq12d 4016 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  x  <->  B  <  A ) )
51, 4orbi12d 793 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  < 
y  \/  y  < 
x )  <->  ( A  <  B  \/  B  < 
A ) ) )
6 df-reap 8530 . . 3  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
75, 6brab2ga 4701 . 2  |-  ( A #  B  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <  B  \/  B  <  A ) ) )
87baib 919 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353    e. wcel 2148   class class class wbr 4003   RRcr 7809    < clt 7990   # creap 8529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-reap 8530
This theorem is referenced by:  reapirr  8532  recexre  8533  reapti  8534  reaplt  8543
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