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Theorem lttri3d 7132
Description: Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
ltd.1  |-  ( ph  ->  A  e.  RR )
ltd.2  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
lttri3d  |-  ( ph  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )

Proof of Theorem lttri3d
StepHypRef Expression
1 ltd.1 . 2  |-  ( ph  ->  A  e.  RR )
2 ltd.2 . 2  |-  ( ph  ->  B  e.  RR )
3 lttri3 7098 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
41, 2, 3syl2anc 391 1  |-  ( ph  ->  ( A  =  B  <-> 
( -.  A  < 
B  /\  -.  B  <  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   class class class wbr 3764   RRcr 6888    < clt 7060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-apti 6999
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065
This theorem is referenced by: (None)
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