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Theorem reapval 7795
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7807 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 3810 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <  y  <->  A  <  B ) )
2 simpr 108 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
3 simpl 107 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
42, 3breq12d 3818 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  x  <->  B  <  A ) )
51, 4orbi12d 740 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  < 
y  \/  y  < 
x )  <->  ( A  <  B  \/  B  < 
A ) ) )
6 df-reap 7794 . . 3  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
75, 6brab2ga 4461 . 2  |-  ( A #  B  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <  B  \/  B  <  A ) ) )
87baib 862 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3805   RRcr 7094    < clt 7267   # creap 7793
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-reap 7794
This theorem is referenced by:  reapirr  7796  recexre  7797  reapti  7798  reaplt  7807
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