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Theorem xrlelttric 24119
Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
Assertion
Ref Expression
xrlelttric  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )

Proof of Theorem xrlelttric
StepHypRef Expression
1 pm2.1 408 . 2  |-  ( -.  B  <  A  \/  B  <  A )
2 xrlenlt 9144 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
32orbi1d 685 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  \/  B  <  A )  <-> 
( -.  B  < 
A  \/  B  < 
A ) ) )
41, 3mpbiri 226 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    e. wcel 1726   class class class wbr 4213   RR*cxr 9120    < clt 9121    <_ cle 9122
This theorem is referenced by:  difioo  24146  esumpcvgval  24469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-le 9127
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