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Theorem reapcotr 8328
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 8318 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  < 
A ) ) )
213adant3 986 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  < 
B  \/  B  < 
A ) ) )
3 axltwlin 7800 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( A  <  C  \/  C  <  B ) ) )
4 axltwlin 7800 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
543com12 1170 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( B  <  C  \/  C  <  A ) ) )
63, 5orim12d 760 . . . 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 702 . . . . 5  |-  ( ( B  <  C  \/  C  <  A )  <->  ( C  <  A  \/  B  < 
C ) )
98orbi2i 736 . . . 4  |-  ( ( ( A  <  C  \/  C  <  B )  \/  ( B  < 
C  \/  C  < 
A ) )  <->  ( ( A  <  C  \/  C  <  B )  \/  ( C  <  A  \/  B  <  C ) ) )
10 or42 746 . . . 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 8318 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  <  C  \/  C  < 
A ) ) )
14133adant2 985 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  C  <->  ( A  < 
C  \/  C  < 
A ) ) )
15 reaplt 8318 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  <  C  \/  C  < 
B ) ) )
16153adant1 984 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B #  C  <->  ( B  < 
C  \/  C  < 
B ) ) )
1714, 16orbi12d 767 . 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 682    /\ w3a 947    e. wcel 1465   class class class wbr 3899   RRcr 7587    < clt 7768   # cap 8311
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312
This theorem is referenced by:  apcotr  8337
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