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Theorem tgbtwnne 24613
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 472 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  G  e. TarskiG )
6 tgbtwntriv2.1 . . . . . 6  |-  ( ph  ->  A  e.  P )
76adantr 472 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  A  e.  P )
8 tgbtwntriv2.2 . . . . . 6  |-  ( ph  ->  B  e.  P )
98adantr 472 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  B  e.  P )
10 tgbtwnne.1 . . . . . . 7  |-  ( ph  ->  B  e.  ( A I C ) )
1110adantr 472 . . . . . 6  |-  ( (
ph  /\  A  =  C )  ->  B  e.  ( A I C ) )
12 simpr 468 . . . . . . 7  |-  ( (
ph  /\  A  =  C )  ->  A  =  C )
1312oveq2d 6324 . . . . . 6  |-  ( (
ph  /\  A  =  C )  ->  ( A I A )  =  ( A I C ) )
1411, 13eleqtrrd 2552 . . . . 5  |-  ( (
ph  /\  A  =  C )  ->  B  e.  ( A I A ) )
151, 2, 3, 5, 7, 9, 14axtgbtwnid 24593 . . . 4  |-  ( (
ph  /\  A  =  C )  ->  A  =  B )
1615eqcomd 2477 . . 3  |-  ( (
ph  /\  A  =  C )  ->  B  =  A )
17 tgbtwnne.2 . . . . 5  |-  ( ph  ->  B  =/=  A )
1817adantr 472 . . . 4  |-  ( (
ph  /\  A  =  C )  ->  B  =/=  A )
1918neneqd 2648 . . 3  |-  ( (
ph  /\  A  =  C )  ->  -.  B  =  A )
2016, 19pm2.65da 586 . 2  |-  ( ph  ->  -.  A  =  C )
2120neqned 2650 1  |-  ( ph  ->  A  =/=  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ` cfv 5589  (class class class)co 6308   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311  df-trkgb 24576  df-trkg 24580
This theorem is referenced by:  mideulem2  24855  opphllem  24856  outpasch  24876  lnopp2hpgb  24884  lmieu  24905  dfcgra2  24950
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