ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrlelttr Unicode version

Theorem xrlelttr 9806
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 529 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <_  B )
2 simpl1 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  e.  RR* )
3 simpl2 1001 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  e.  RR* )
4 xrlenlt 8022 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
52, 3, 4syl2anc 411 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
61, 5mpbid 147 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  -.  B  <  A )
76pm2.21d 619 . . 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 531 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  B  <  C )
10 simpl3 1002 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  C  e.  RR* )
11 xrltso 9796 . . . . . 6  |-  <  Or  RR*
12 sowlin 4321 . . . . . 6  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* ) )  -> 
( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
1311, 12mpan 424 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  ( B  <  C  ->  ( B  <  A  \/  A  <  C ) ) )
143, 10, 2, 13syl3anc 1238 . . . 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 718 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <  C ) )  ->  A  <  C )
1716ex 115 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 104    <-> wb 105    \/ wo 708    /\ w3a 978    e. wcel 2148   class class class wbr 4004    Or wor 4296   RR*cxr 7991    < clt 7992    <_ cle 7993
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925
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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-po 4297  df-iso 4298  df-xp 4633  df-cnv 4635  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998
This theorem is referenced by:  xrlelttrd  9810  xrre  9820  xrre2  9821  iooss1  9916  iccssioo  9942  iccssico  9945  iocssioo  9963  ioossioo  9965  ico0  10262  bldisj  13904  xblm  13920  blsscls2  13996  metcnpi3  14020
  Copyright terms: Public domain W3C validator