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Theorem reapval 8306
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8318 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 3904 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <  y  <->  A  <  B ) )
2 simpr 109 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
3 simpl 108 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
42, 3breq12d 3912 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  x  <->  B  <  A ) )
51, 4orbi12d 767 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  < 
y  \/  y  < 
x )  <->  ( A  <  B  \/  B  < 
A ) ) )
6 df-reap 8305 . . 3  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
75, 6brab2ga 4584 . 2  |-  ( A #  B  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <  B  \/  B  <  A ) ) )
87baib 889 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   class class class wbr 3899   RRcr 7587    < clt 7768   # creap 8304
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-reap 8305
This theorem is referenced by:  reapirr  8307  recexre  8308  reapti  8309  reaplt  8318
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