btwnlng2 - Metamath Proof Explorer

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Theorem btwnlng2 26408

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)
btwnlng2.1	⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))

**Assertion**
Ref	Expression
btwnlng2	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

**Proof of Theorem btwnlng2**
Step	Hyp	Ref	Expression
1		btwnlng2.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))
2	1	3mix2d 1333	. 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 26341	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
12	2, 11	mpbird 259	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-trkg 26241

This theorem is referenced by: mirln 26464 colperpexlem3 26520 outpasch 26543 hpgerlem 26553

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