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Theorem axltwlin 7999
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7899 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
Assertion
Ref Expression
axltwlin  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )

Proof of Theorem axltwlin
StepHypRef Expression
1 ax-pre-ltwlin 7899 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
2 ltxrlt 7997 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 1017 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 ltxrlt 7997 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A 
<RR  C ) )
543adant2 1016 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A  <RR  C ) )
6 ltxrlt 7997 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  C 
<RR  B ) )
76ancoms 268 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  C 
<RR  B ) )
873adant1 1015 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  C  <RR  B ) )
95, 8orbi12d 793 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  \/  C  <  B )  <-> 
( A  <RR  C  \/  C  <RR  B ) ) )
101, 3, 93imtr4d 203 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 708    /\ w3a 978    e. wcel 2146   class class class wbr 3998   RRcr 7785    <RR cltrr 7790    < clt 7966
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-ltwlin 7899
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-pnf 7968  df-mnf 7969  df-ltxr 7971
This theorem is referenced by:  ltso  8009  letr  8014  lelttr  8020  ltletr  8021  gt0add  8504  reapcotr  8529  sup3exmid  8885  xrltso  9765  rebtwn2zlemstep  10221  expnbnd  10611  leabs  11050  ltabs  11063  abslt  11064  absle  11065  maxabslemlub  11183  suplociccreex  13595  ivthinclemloc  13612  cnplimclemle  13630  reeff1o  13687  efltlemlt  13688  sin0pilem2  13696  coseq0negpitopi  13750  cos02pilt1  13765
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