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Theorem xrlelttr 9557
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 505 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 969 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR* )
3 simpl2 970 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR* )
4 xrlenlt 7797 . . . . . 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 593 . . 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 506 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 971 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR* )
11 xrltso 9550 . . . . . 6  |-  <  Or  RR*
12 sowlin 4212 . . . . . 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 1201 . . . 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 692 . 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 682    /\ w3a 947    e. wcel 1465   class class class wbr 3899    Or wor 4187   RR*cxr 7767    < clt 7768    <_ cle 7769
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-po 4188  df-iso 4189  df-xp 4515  df-cnv 4517  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774
This theorem is referenced by:  xrlelttrd  9561  xrre  9571  xrre2  9572  iooss1  9667  iccssioo  9693  iccssico  9696  iocssioo  9714  ioossioo  9716  ico0  10007  bldisj  12497  xblm  12513  blsscls2  12589  metcnpi3  12613
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