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Theorem ltleletr 8013
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 8005 . . . . . 6  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
213coml 1210 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <  A  /\  A  <  B )  ->  C  <  B
) )
32expcomd 1439 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  <  A  ->  C  <  B ) ) )
4 con3 642 . . . 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 8007 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
763adant1 1015 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
8 lenlt 8007 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
983adant2 1016 . . . 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 978    e. wcel 2146   class class class wbr 3998   RRcr 7785    < clt 7966    <_ cle 7967
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-pre-lttrn 7900
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972
This theorem is referenced by:  nn0ge2m1nn  9207  lbzbi  9587  iseqf1olemqk  10462
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