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Theorem reapval 8495
Description: Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8507 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 3994 . . . 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 4002 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  x  <->  B  <  A ) )
51, 4orbi12d 788 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  < 
y  \/  y  < 
x )  <->  ( A  <  B  \/  B  < 
A ) ) )
6 df-reap 8494 . . 3  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
75, 6brab2ga 4686 . 2  |-  ( A #  B  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <  B  \/  B  <  A ) ) )
87baib 914 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 703    = wceq 1348    e. wcel 2141   class class class wbr 3989   RRcr 7773    < clt 7954   # creap 8493
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-reap 8494
This theorem is referenced by:  reapirr  8496  recexre  8497  reapti  8498  reaplt  8507
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