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Theorem reapti 8056
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8099. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
Assertion
Ref Expression
reapti  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <->  -.  A #  B ) )

Proof of Theorem reapti
StepHypRef Expression
1 ltnr 7562 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
21adantr 270 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  <  A
)
3 oridm 709 . . . . . 6  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
4 breq2 3849 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
5 breq1 3848 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  B  <  A ) )
64, 5orbi12d 742 . . . . . 6  |-  ( A  =  B  ->  (
( A  <  A  \/  A  <  A )  <-> 
( A  <  B  \/  B  <  A ) ) )
73, 6syl5bbr 192 . . . . 5  |-  ( A  =  B  ->  ( A  <  A  <->  ( A  <  B  \/  B  < 
A ) ) )
87notbid 627 . . . 4  |-  ( A  =  B  ->  ( -.  A  <  A  <->  -.  ( A  <  B  \/  B  <  A ) ) )
92, 8syl5ibcom 153 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  ( A  <  B  \/  B  < 
A ) ) )
10 reapval 8053 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
1110notbid 627 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
129, 11sylibrd 167 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  A #  B )
)
13 axapti 7557 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
14133expia 1145 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  ->  A  =  B ) )
1511, 14sylbid 148 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  ->  A  =  B )
)
1612, 15impbid 127 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <->  -.  A #  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   class class class wbr 3845   RRcr 7349    < clt 7522   # creap 8051
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-pre-ltirr 7457  ax-pre-apti 7460
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7524  df-mnf 7525  df-ltxr 7527  df-reap 8052
This theorem is referenced by:  rimul  8062  apreap  8064  apti  8099
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