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Theorem axapti 7828
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7728 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
axapti  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )

Proof of Theorem axapti
StepHypRef Expression
1 ltxrlt 7823 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
2 ltxrlt 7823 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
32ancoms 266 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  A  <->  B 
<RR  A ) )
41, 3orbi12d 782 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  < 
B  \/  B  < 
A )  <->  ( A  <RR  B  \/  B  <RR  A ) ) )
54notbid 656 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  <->  -.  ( A  <RR  B  \/  B  <RR  A ) ) )
6 ax-pre-apti 7728 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )
763expia 1183 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A 
<RR  B  \/  B  <RR  A )  ->  A  =  B ) )
85, 7sylbid 149 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -.  ( A  <  B  \/  B  <  A )  ->  A  =  B ) )
983impia 1178 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  -.  ( A  <  B  \/  B  <  A ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924   RRcr 7612    <RR cltrr 7617    < clt 7793
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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-apti 7728
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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-ltxr 7798
This theorem is referenced by:  lttri3  7837  reapti  8334
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