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

Theorem reapcotr 8557
Description: Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
Assertion
Ref Expression
reapcotr  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )

Proof of Theorem reapcotr
StepHypRef Expression
1 reaplt 8547 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
213adant3 1017 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
3 axltwlin 8027 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
4 axltwlin 8027 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
543com12 1207 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
63, 5orim12d 786 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  \/  B  <  A )  ->  ( ( A  <  C  \/  C  <  B )  \/  ( B  <  C  \/  C  <  A ) ) ) )
72, 6sylbid 150 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( ( A  <  C  \/  C  <  B )  \/  ( B  <  C  \/  C  <  A ) ) ) )
8 orcom 728 . . . . 5  |-  ( ( B  <  C  \/  C  <  A )  <->  ( C  <  A  \/  B  < 
C ) )
98orbi2i 762 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  B )  \/  ( C  <  A  \/  B  <  C ) ) )
10 or42 772 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( C  < 
A  \/  B  < 
C ) )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) )
119, 10bitri 184 . . 3  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) )
127, 11imbitrdi 161 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) ) )
13 reaplt 8547 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  <  C  \/  C  < 
A ) ) )
14133adant2 1016 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  < 
C  \/  C  < 
A ) ) )
15 reaplt 8547 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  <  C  \/  C  < 
B ) ) )
16153adant1 1015 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  < 
C  \/  C  < 
B ) ) )
1714, 16orbi12d 793 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A #  C  \/  B #  C )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) ) )
1812, 17sylibrd 169 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( A #  C  \/  B #  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 708    /\ w3a 978    e. wcel 2148   class class class wbr 4005   RRcr 7812    < clt 7994   # cap 8540
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930
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-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541
This theorem is referenced by:  apcotr  8566
  Copyright terms: Public domain W3C validator