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Theorem tgbtwntriv1 24083
Description: Betweeness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-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 )
Assertion
Ref Expression
tgbtwntriv1  |-  ( ph  ->  A  e.  ( A I B ) )

Proof of Theorem tgbtwntriv1
StepHypRef Expression
1 tkgeom.p . 2  |-  P  =  ( Base `  G
)
2 tkgeom.d . 2  |-  .-  =  ( dist `  G )
3 tkgeom.i . 2  |-  I  =  (Itv `  G )
4 tkgeom.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 tgbtwntriv2.2 . 2  |-  ( ph  ->  B  e.  P )
6 tgbtwntriv2.1 . 2  |-  ( ph  ->  A  e.  P )
71, 2, 3, 4, 5, 6tgbtwntriv2 24079 . 2  |-  ( ph  ->  A  e.  ( B I A ) )
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 24080 1  |-  ( ph  ->  A  e.  ( A I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048
This theorem is referenced by:  tgldim0itv  24096  legtri3  24178  leg0  24180  legbtwn  24182  ncolne1  24206  tglnne  24209  tglinerflx1  24214  mirinv  24248  miriso  24251  colmid  24266  krippenlem  24268  colperpex  24308
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