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Theorem reapval 8452
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8464 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 3971 . . . 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 3979 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  x  <->  B  <  A ) )
51, 4orbi12d 783 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  < 
y  \/  y  < 
x )  <->  ( A  <  B  \/  B  < 
A ) ) )
6 df-reap 8451 . . 3  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
75, 6brab2ga 4662 . 2  |-  ( A #  B  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <  B  \/  B  <  A ) ) )
87baib 905 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 698    = wceq 1335    e. wcel 2128   class class class wbr 3966   RRcr 7732    < clt 7913   # creap 8450
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-reap 8451
This theorem is referenced by:  reapirr  8453  recexre  8454  reapti  8455  reaplt  8464
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