ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reapti Unicode version

Theorem reapti 8341
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8384. (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 7841 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
21adantr 274 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  <  A
)
3 oridm 746 . . . . . 6  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
4 breq2 3933 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
5 breq1 3932 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  B  <  A ) )
64, 5orbi12d 782 . . . . . 6  |-  ( A  =  B  ->  (
( A  <  A  \/  A  <  A )  <-> 
( A  <  B  \/  B  <  A ) ) )
73, 6syl5bbr 193 . . . . 5  |-  ( A  =  B  ->  ( A  <  A  <->  ( A  <  B  \/  B  < 
A ) ) )
87notbid 656 . . . 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 8338 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
1110notbid 656 . . 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 7835 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
14133expia 1183 . . 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 697    = wceq 1331    e. wcel 1480   class class class wbr 3929   RRcr 7619    < clt 7800   # creap 8336
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-apti 7735
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-reap 8337
This theorem is referenced by:  rimul  8347  apreap  8349  apti  8384
  Copyright terms: Public domain W3C validator