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Theorem xrlelttr 8823
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 491 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 918 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR* )
3 simpl2 919 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR* )
4 xrlenlt 7143 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 397 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 139 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 559 . . 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 492 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 920 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR* )
11 xrltso 8818 . . . . . 6  |-  <  Or  RR*
12 sowlin 4085 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* ) )  -> 
( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
1311, 12mpan 408 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
143, 10, 2, 13syl3anc 1146 . . . 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 648 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1716ex 112 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   class class class wbr 3792    Or wor 4060   RR*cxr 7118    < clt 7119    <_ cle 7120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-po 4061  df-iso 4062  df-xp 4379  df-cnv 4381  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125
This theorem is referenced by:  xrlelttrd  8827  xrre  8834  xrre2  8835  iooss1  8886  iccssioo  8912  iccssico  8915  iocssioo  8933  ioossioo  8935  ico0  9218
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