Theorem btwnlng1 28672

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 1338	. 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 28606	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
12	2, 11	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  TarskiGcstrkg 28480  Itvcitv 28486  LineGclng 28487

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6447  df-fun 6493  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-trkg 28506

This theorem is referenced by:  tglnne  28681  tglinerflx1  28686  tglinerflx2  28687  coltr3  28701  mirln2  28730  midexlem  28745  colperpexlem3  28785  mideulem2  28787  opphllem1  28800  opphllem2  28801  opphllem4  28803  hlpasch  28809  lnopp2hpgb  28816  colopp  28822  lmieu  28837  lmimid  28847  lmiisolem  28849  hypcgrlem1  28852  hypcgrlem2  28853  trgcopyeulem  28858