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Theorem xrlelttric 23265
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 pm2.1 406 . 2  |-  ( -.  B  <  A  \/  B  <  A )
2 xrlenlt 8906 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
32orbi1d 683 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <_  B  \/  B  <  A )  <-> 
( -.  B  < 
A  \/  B  < 
A ) ) )
41, 3mpbiri 224 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1696   class class class wbr 4039   RR*cxr 8882    < clt 8883    <_ cle 8884
This theorem is referenced by:  difioo  23290  esumpcvgval  23461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-le 8889
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