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Theorem xrletrid 9792
Description: Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
xrletrid.1  |-  ( ph  ->  A  e.  RR* )
xrletrid.2  |-  ( ph  ->  B  e.  RR* )
xrletrid.3  |-  ( ph  ->  A  <_  B )
xrletrid.4  |-  ( ph  ->  B  <_  A )
Assertion
Ref Expression
xrletrid  |-  ( ph  ->  A  =  B )

Proof of Theorem xrletrid
StepHypRef Expression
1 xrletrid.3 . 2  |-  ( ph  ->  A  <_  B )
2 xrletrid.4 . 2  |-  ( ph  ->  B  <_  A )
3 xrletrid.1 . . 3  |-  ( ph  ->  A  e.  RR* )
4 xrletrid.2 . . 3  |-  ( ph  ->  B  e.  RR* )
5 xrletri3 9791 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) ) )
63, 4, 5syl2anc 411 . 2  |-  ( ph  ->  ( A  =  B  <-> 
( A  <_  B  /\  B  <_  A ) ) )
71, 2, 6mpbir2and 944 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4000   RR*cxr 7981    <_ cle 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914  ax-pre-apti 7917
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988
This theorem is referenced by: (None)
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