ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axltwlin Unicode version

Theorem axltwlin 7554
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7458 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 7458 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
2 ltxrlt 7552 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 963 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 ltxrlt 7552 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A 
<RR  C ) )
543adant2 962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  A  <RR  C ) )
6 ltxrlt 7552 . . . . 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 961 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  C  <RR  B ) )
95, 8orbi12d 742 . 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 664    /\ w3a 924    e. wcel 1438   class class class wbr 3845   RRcr 7349    <RR cltrr 7354    < clt 7522
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-pre-ltwlin 7458
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7524  df-mnf 7525  df-ltxr 7527
This theorem is referenced by:  ltso  7563  letr  7568  lelttr  7573  ltletr  7574  gt0add  8050  reapcotr  8075  xrltso  9266  rebtwn2zlemstep  9664  expnbnd  10077  leabs  10507  ltabs  10520  abslt  10521  absle  10522  maxabslemlub  10640
  Copyright terms: Public domain W3C validator