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Theorem reapti 7644
Description: Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7687. (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 7154 . . . . 5  |-  ( A  e.  RR  ->  -.  A  <  A )
21adantr 265 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A  <  A
)
3 oridm 684 . . . . . 6  |-  ( ( A  <  A  \/  A  <  A )  <->  A  <  A )
4 breq2 3796 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  A  <  B ) )
5 breq1 3795 . . . . . . 7  |-  ( A  =  B  ->  ( A  <  A  <->  B  <  A ) )
64, 5orbi12d 717 . . . . . 6  |-  ( A  =  B  ->  (
( A  <  A  \/  A  <  A )  <-> 
( A  <  B  \/  B  <  A ) ) )
73, 6syl5bbr 187 . . . . 5  |-  ( A  =  B  ->  ( A  <  A  <->  ( A  <  B  \/  B  < 
A ) ) )
87notbid 602 . . . 4  |-  ( A  =  B  ->  ( -.  A  <  A  <->  -.  ( A  <  B  \/  B  <  A ) ) )
92, 8syl5ibcom 148 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  ( A  <  B  \/  B  < 
A ) ) )
10 reapval 7641 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
1110notbid 602 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  <->  -.  ( A  <  B  \/  B  <  A ) ) )
129, 11sylibrd 162 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  ->  -.  A #  B )
)
13 axapti 7149 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
14133expia 1117 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  ->  A  =  B ) )
1511, 14sylbid 143 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  A #  B  ->  A  =  B )
)
1612, 15impbid 124 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <->  -.  A #  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   class class class wbr 3792   RRcr 6946    < clt 7119   # creap 7639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-apti 7057
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-pnf 7121  df-mnf 7122  df-ltxr 7124  df-reap 7640
This theorem is referenced by:  rimul  7650  apreap  7652  apti  7687
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