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Theorem xrlelttr 9240
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 498 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 946 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR* )
3 simpl2 947 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR* )
4 xrlenlt 7530 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 403 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 145 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 584 . . 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 499 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 948 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR* )
11 xrltso 9235 . . . . . 6  |-  <  Or  RR*
12 sowlin 4138 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* ) )  -> 
( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
1311, 12mpan 415 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
143, 10, 2, 13syl3anc 1174 . . . 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 673 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1716ex 113 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    e. wcel 1438   class class class wbr 3837    Or wor 4113   RR*cxr 7500    < clt 7501    <_ cle 7502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-po 4114  df-iso 4115  df-xp 4434  df-cnv 4436  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507
This theorem is referenced by:  xrlelttrd  9244  xrre  9251  xrre2  9252  iooss1  9303  iccssioo  9329  iccssico  9332  iocssioo  9350  ioossioo  9352  ico0  9638
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