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Theorem xrlttrd 9841
Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
xrlttrd.1  |-  ( ph  ->  A  e.  RR* )
xrlttrd.2  |-  ( ph  ->  B  e.  RR* )
xrlttrd.3  |-  ( ph  ->  C  e.  RR* )
xrlttrd.4  |-  ( ph  ->  A  <  B )
xrlttrd.5  |-  ( ph  ->  B  <  C )
Assertion
Ref Expression
xrlttrd  |-  ( ph  ->  A  <  C )

Proof of Theorem xrlttrd
StepHypRef Expression
1 xrlttrd.4 . 2  |-  ( ph  ->  A  <  B )
2 xrlttrd.5 . 2  |-  ( ph  ->  B  <  C )
3 xrlttrd.1 . . 3  |-  ( ph  ->  A  e.  RR* )
4 xrlttrd.2 . . 3  |-  ( ph  ->  B  e.  RR* )
5 xrlttrd.3 . . 3  |-  ( ph  ->  C  e.  RR* )
6 xrlttr 9827 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  B  <  C )  ->  A  <  C
) )
73, 4, 5, 6syl3anc 1249 . 2  |-  ( ph  ->  ( ( A  < 
B  /\  B  <  C )  ->  A  <  C ) )
81, 2, 7mp2and 433 1  |-  ( ph  ->  A  <  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   class class class wbr 4018   RR*cxr 8022    < clt 8023
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-pre-lttrn 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028
This theorem is referenced by:  xlt2add  9912  ioom  10293
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