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Theorem reapcotr 8136
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 8126 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
213adant3 964 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
3 axltwlin 7615 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
4 axltwlin 7615 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
543com12 1148 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
63, 5orim12d 736 . . . 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 683 . . . . 5  |-  ( ( B  <  C  \/  C  <  A )  <->  ( C  <  A  \/  B  < 
C ) )
98orbi2i 715 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  B )  \/  ( C  <  A  \/  B  <  C ) ) )
10 or42 725 . . . 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 8126 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  <  C  \/  C  < 
A ) ) )
14133adant2 963 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  < 
C  \/  C  < 
A ) ) )
15 reaplt 8126 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  <  C  \/  C  < 
B ) ) )
16153adant1 962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  < 
C  \/  C  < 
B ) ) )
1714, 16orbi12d 743 . 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 665    /\ w3a 925    e. wcel 1439   class class class wbr 3851   RRcr 7410    < clt 7583   # cap 8119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-ltxr 7588  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120
This theorem is referenced by:  apcotr  8145
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