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Theorem lelttrdi 8381
Description: If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
Hypotheses
Ref Expression
lelttrdi.r  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
)
lelttrdi.l  |-  ( ph  ->  B  <_  C )
Assertion
Ref Expression
lelttrdi  |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )

Proof of Theorem lelttrdi
StepHypRef Expression
1 lelttrdi.r . . . . 5  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
)
21simp1d 1009 . . . 4  |-  ( ph  ->  A  e.  RR )
32adantr 276 . . 3  |-  ( (
ph  /\  A  <  B )  ->  A  e.  RR )
41simp2d 1010 . . . 4  |-  ( ph  ->  B  e.  RR )
54adantr 276 . . 3  |-  ( (
ph  /\  A  <  B )  ->  B  e.  RR )
61simp3d 1011 . . . 4  |-  ( ph  ->  C  e.  RR )
76adantr 276 . . 3  |-  ( (
ph  /\  A  <  B )  ->  C  e.  RR )
8 simpr 110 . . 3  |-  ( (
ph  /\  A  <  B )  ->  A  <  B )
9 lelttrdi.l . . . 4  |-  ( ph  ->  B  <_  C )
109adantr 276 . . 3  |-  ( (
ph  /\  A  <  B )  ->  B  <_  C )
113, 5, 7, 8, 10ltletrd 8378 . 2  |-  ( (
ph  /\  A  <  B )  ->  A  <  C )
1211ex 115 1  |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    e. wcel 2148   class class class wbr 4003   RRcr 7809    < clt 7990    <_ cle 7991
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltwlin 7923
This theorem depends on definitions:  df-bi 117  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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996
This theorem is referenced by:  difgtsumgt  9320  subfzo0  10239
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