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Theorem xrlelttr 9589
Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
xrlelttr  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <  C )  ->  A  <  C
) )

Proof of Theorem xrlelttr
StepHypRef Expression
1 simprl 520 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 984 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR* )
3 simpl2 985 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR* )
4 xrlenlt 7829 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 408 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 146 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 608 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( B  <  A  ->  A  <  C ) )
8 idd 21 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( A  <  C  ->  A  <  C ) )
9 simprr 521 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 986 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR* )
11 xrltso 9582 . . . . . 6  |-  <  Or  RR*
12 sowlin 4242 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* ) )  -> 
( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
1311, 12mpan 420 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
143, 10, 2, 13syl3anc 1216 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
159, 14mpd 13 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( B  <  A  \/  A  <  C ) )
167, 8, 15mpjaod 707 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1716ex 114 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <_  B  /\  B  <  C )  ->  A  <  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    e. wcel 1480   class class class wbr 3929    Or wor 4217   RR*cxr 7799    < clt 7800    <_ cle 7801
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-po 4218  df-iso 4219  df-xp 4545  df-cnv 4547  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806
This theorem is referenced by:  xrlelttrd  9593  xrre  9603  xrre2  9604  iooss1  9699  iccssioo  9725  iccssico  9728  iocssioo  9746  ioossioo  9748  ico0  10039  bldisj  12570  xblm  12586  blsscls2  12662  metcnpi3  12686
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