Theorem btwnlng1 28868

Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)

Hypotheses

Ref	Expression
btwnlng1.p	⊢ 𝑃 = (Base‘𝐺)
btwnlng1.i	⊢ 𝐼 = (Itv‘𝐺)
btwnlng1.l	⊢ 𝐿 = (LineG‘𝐺)
btwnlng1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
btwnlng1.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
btwnlng1.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
btwnlng1.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwnlng1.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)
btwnlng1.1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌))

Assertion

Ref	Expression
btwnlng1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng1

Step	Hyp	Ref	Expression
1		btwnlng1.1	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌))
2	1	3mix1d 1353	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
3		btwnlng1.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
4		btwnlng1.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		btwnlng1.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
6		btwnlng1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		btwnlng1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		btwnlng1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9		btwnlng1.d	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
10		btwnlng1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	3, 4, 5, 6, 7, 8, 9, 10	tgellng 28798	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
12	2, 11	mpbird 260	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1100   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  TarskiGcstrkg 28672  Itvcitv 28678  LineGclng 28679

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-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  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-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-trkg 28698

This theorem is referenced by:  tglnne  28877  tglinerflx1  28882  tglinerflx2  28883  coltr3  28898  tglnpt3  28903  mirln2  28930  midexlem  28945  colperpexlem3  28988  mideulem2  28990  opphllem1  29003  opphllem2  29004  opphllem4  29006  hlpasch  29013  lnopp2hpgb  29020  colopp  29026  plngrotlem1  29043  lmieu  29067  lmimid  29077  lmiisolem  29079  hypcgrlem1  29082  hypcgrlem2  29083  trgcopyeulem  29089  prlnghpg  29169  prlngmolem1  29175  prlngmid2  29183  btwnlng13  35023