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Theorem ltleletr 8070
Description: Transitive law, weaker form of  ( A  < 
B  /\  B  <_  C )  ->  A  <  C. (Contributed by AV, 14-Oct-2018.)
Assertion
Ref Expression
ltleletr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  /\  B  <_  C )  ->  A  <_  C
) )

Proof of Theorem ltleletr
StepHypRef Expression
1 lttr 8062 . . . . . 6  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
213coml 1212 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
32expcomd 1452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  <  A  ->  C  <  B ) ) )
4 con3 643 . . . 4  |-  ( ( C  <  A  ->  C  <  B )  -> 
( -.  C  < 
B  ->  -.  C  <  A ) )
53, 4syl6 33 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( -.  C  <  B  ->  -.  C  <  A ) ) )
6 lenlt 8064 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
763adant1 1017 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
8 lenlt 8064 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
983adant2 1018 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
107, 9imbi12d 234 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( B  <_  C  ->  A  <_  C )  <->  ( -.  C  <  B  ->  -.  C  <  A
) ) )
115, 10sylibrd 169 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( B  <_  C  ->  A  <_  C ) ) )
1211impd 254 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    /\ w3a 980    e. wcel 2160   class class class wbr 4018   RRcr 7841    < clt 8023    <_ cle 8024
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-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-cnv 4652  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029
This theorem is referenced by:  nn0ge2m1nn  9267  lbzbi  9648  iseqf1olemqk  10527
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