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Theorem reapcotr 8372
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 8362 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
213adant3 1001 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
3 axltwlin 7844 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
4 axltwlin 7844 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
543com12 1185 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
63, 5orim12d 775 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  B  \/  B  <  A )  ->  ( ( A  <  C  \/  C  <  B )  \/  ( B  <  C  \/  C  <  A ) ) ) )
72, 6sylbid 149 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( ( A  <  C  \/  C  <  B )  \/  ( B  <  C  \/  C  <  A ) ) ) )
8 orcom 717 . . . . 5  |-  ( ( B  <  C  \/  C  <  A )  <->  ( C  <  A  \/  B  < 
C ) )
98orbi2i 751 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  B )  \/  ( C  <  A  \/  B  <  C ) ) )
10 or42 761 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( C  < 
A  \/  B  < 
C ) )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) )
119, 10bitri 183 . . 3  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) )
127, 11syl6ib 160 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) ) )
13 reaplt 8362 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  <  C  \/  C  < 
A ) ) )
14133adant2 1000 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  < 
C  \/  C  < 
A ) ) )
15 reaplt 8362 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  <  C  \/  C  < 
B ) ) )
16153adant1 999 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  < 
C  \/  C  < 
B ) ) )
1714, 16orbi12d 782 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A #  C  \/  B #  C )  <->  ( ( A  <  C  \/  C  <  A )  \/  ( B  <  C  \/  C  <  B ) ) ) )
1812, 17sylibrd 168 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 104    \/ wo 697    /\ w3a 962    e. wcel 1480   class class class wbr 3929   RRcr 7631    < clt 7812   # cap 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-ltxr 7817  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356
This theorem is referenced by:  apcotr  8381
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