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Theorem axltwlin 7236
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7140 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 7140 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
2 ltxrlt 7234 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 959 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 ltxrlt 7234 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A 
<RR  C ) )
543adant2 958 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A  <RR  C ) )
6 ltxrlt 7234 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  C 
<RR  B ) )
76ancoms 264 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  C 
<RR  B ) )
873adant1 957 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  C  <RR  B ) )
95, 8orbi12d 740 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  \/  C  <  B )  <-> 
( A  <RR  C  \/  C  <RR  B ) ) )
101, 3, 93imtr4d 201 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 103    \/ wo 662    /\ w3a 920    e. wcel 1434   class class class wbr 3787   RRcr 7031    <RR cltrr 7036    < clt 7204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-pre-ltwlin 7140
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-xp 4371  df-pnf 7206  df-mnf 7207  df-ltxr 7209
This theorem is referenced by:  ltso  7245  letr  7250  lelttr  7255  ltletr  7256  gt0add  7729  reapcotr  7754  xrltso  8936  rebtwn2zlemstep  9328  expnbnd  9682  leabs  10087  ltabs  10100  abslt  10101  absle  10102  maxabslemlub  10220
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