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Theorem reapti 8309
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8352. (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 7809 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
21adantr 274 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  <  A
)
3 oridm 731 . . . . . 6  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
4 breq2 3903 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
5 breq1 3902 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  B  <  A ) )
64, 5orbi12d 767 . . . . . 6  |-  ( A  =  B  ->  (
( A  <  A  \/  A  <  A )  <-> 
( A  <  B  \/  B  <  A ) ) )
73, 6syl5bbr 193 . . . . 5  |-  ( A  =  B  ->  ( A  <  A  <->  ( A  <  B  \/  B  < 
A ) ) )
87notbid 641 . . . 4  |-  ( A  =  B  ->  ( -.  A  <  A  <->  -.  ( A  <  B  \/  B  <  A ) ) )
92, 8syl5ibcom 154 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  ( A  <  B  \/  B  < 
A ) ) )
10 reapval 8306 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
1110notbid 641 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
129, 11sylibrd 168 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  A #  B )
)
13 axapti 7803 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
14133expia 1168 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  ->  A  =  B ) )
1511, 14sylbid 149 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  ->  A  =  B )
)
1612, 15impbid 128 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <->  -.  A #  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> 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-in1 588  ax-in2 589  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-13 1476  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  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700  ax-pre-apti 7703
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  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-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-reap 8305
This theorem is referenced by:  rimul  8315  apreap  8317  apti  8352
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