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Theorem tgbtwntriv1 27380
Description: Betweenness 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 (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))

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 (𝜑𝐺 ∈ TarskiG)
5 tgbtwntriv2.2 . 2 (𝜑𝐵𝑃)
6 tgbtwntriv2.1 . 2 (𝜑𝐴𝑃)
71, 2, 3, 4, 5, 6tgbtwntriv2 27376 . 2 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 27377 1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7356  Basecbs 17082  distcds 17141  TarskiGcstrkg 27316  Itvcitv 27322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7359  df-trkgc 27337  df-trkgb 27338  df-trkgcb 27339  df-trkg 27342
This theorem is referenced by:  tgldim0itv  27393  legtri3  27479  leg0  27481  legbtwn  27483  ncolne1  27514  tglnne  27517  tglinerflx1  27522  mirinv  27555  miriso  27559  colmid  27577  krippenlem  27579  colperpex  27622  outpasch  27644  hlpasch  27645
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