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Theorem tgbtwntriv1 23603
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 23599 . 2  |-  ( ph  ->  A  e.  ( B I A ) )
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 23600 1  |-  ( ph  ->  A  e.  ( A I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571
This theorem is referenced by:  tgldim0itv  23616  legtri3  23697  leg0  23699  legbtwn  23701  ncolne1  23712  tglnne  23715  tglinerflx1  23720  tglnpt2  23727  tglineneq  23730  mirinv  23753  miriso  23756  colmid  23766  krippenlem  23768  colperpex  23805
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