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Theorem tgbtwnne 23750
Description: Betweenness and inequality (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
tkgeom.p  |-  P  =  ( Base `  G
)
tkgeom.d  |-  .-  =  ( dist `  G )
tkgeom.i  |-  I  =  (Itv `  G )
tkgeom.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwntriv2.1  |-  ( ph  ->  A  e.  P )
tgbtwntriv2.2  |-  ( ph  ->  B  e.  P )
tgbtwncomb.3  |-  ( ph  ->  C  e.  P )
tgbtwnne.1  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnne.2  |-  ( ph  ->  B  =/=  A )
Assertion
Ref Expression
tgbtwnne  |-  ( ph  ->  A  =/=  C )

Proof of Theorem tgbtwnne
StepHypRef Expression
1 tkgeom.p . . . . 5  |-  P  =  ( Base `  G
)
2 tkgeom.d . . . . 5  |-  .-  =  ( dist `  G )
3 tkgeom.i . . . . 5  |-  I  =  (Itv `  G )
4 tkgeom.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  G  e. TarskiG )
6 tgbtwntriv2.1 . . . . . 6  |-  ( ph  ->  A  e.  P )
76adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  A  e.  P )
8 tgbtwntriv2.2 . . . . . 6  |-  ( ph  ->  B  e.  P )
98adantr 465 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  B  e.  P )
10 tgbtwnne.1 . . . . . . 7  |-  ( ph  ->  B  e.  ( A I C ) )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  A  =  C )  ->  B  e.  ( A I C ) )
12 simpr 461 . . . . . . 7  |-  ( (
ph  /\  A  =  C )  ->  A  =  C )
1312oveq2d 6294 . . . . . 6  |-  ( (
ph  /\  A  =  C )  ->  ( A I A )  =  ( A I C ) )
1411, 13eleqtrrd 2532 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  B  e.  ( A I A ) )
151, 2, 3, 5, 7, 9, 14axtgbtwnid 23732 . . . 4  |-  ( (
ph  /\  A  =  C )  ->  A  =  B )
1615eqcomd 2449 . . 3  |-  ( (
ph  /\  A  =  C )  ->  B  =  A )
17 tgbtwnne.2 . . . . 5  |-  ( ph  ->  B  =/=  A )
1817adantr 465 . . . 4  |-  ( (
ph  /\  A  =  C )  ->  B  =/=  A )
1918neneqd 2643 . . 3  |-  ( (
ph  /\  A  =  C )  ->  -.  B  =  A )
2016, 19pm2.65da 576 . 2  |-  ( ph  ->  -.  A  =  C )
2120neqned 2644 1  |-  ( ph  ->  A  =/=  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   ` cfv 5575  (class class class)co 6278   Basecbs 14506   distcds 14580  TarskiGcstrkg 23694  Itvcitv 23701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-iota 5538  df-fv 5583  df-ov 6281  df-trkgb 23714  df-trkg 23719
This theorem is referenced by:  mideulem2  23977  opphllem  23978  lnopp2hpgb  24001  lmieu  24019
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