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Theorem xrlelttric 27722
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 pm2.1 417 . 2  |-  ( -.  B  <  A  \/  B  <  A )
2 xrlenlt 9669 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
32orbi1d 702 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  \/  B  <  A )  <-> 
( -.  B  < 
A  \/  B  < 
A ) ) )
41, 3mpbiri 233 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    e. wcel 1819   class class class wbr 4456   RR*cxr 9644    < clt 9645    <_ cle 9646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-le 9651
This theorem is referenced by:  difioo  27745  esumpcvgval  28240
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