Theorem tgbtwntriv1 28736

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
tgbtwntriv1	⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv1

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwntriv2.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
6		tgbtwntriv2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	1, 2, 3, 4, 5, 6	tgbtwntriv2 28732	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
8	1, 2, 3, 4, 5, 6, 6, 7	tgbtwncom 28733	1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  distcds 17318  TarskiGcstrkg 28672  Itvcitv 28678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7413  df-trkgc 28693  df-trkgb 28694  df-trkgcb 28695  df-trkg 28698

This theorem is referenced by:  tgldim0itv  28749  legtri3  28835  leg0  28837  legbtwn  28839  ncolne1  28874  tglnne  28877  tglinerflx1  28882  mirinv  28919  miriso  28923  colmid  28941  krippenlem  28943  colperpex  28989  outpasch  29012  hlpasch  29013