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Theorem btwncolg2 28501

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

**Hypotheses**
Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwncolg2.z	⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))

**Assertion**
Ref	Expression
btwncolg2	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

**Proof of Theorem btwncolg2**
Step	Hyp	Ref	Expression
1		btwncolg2.z	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))
2	1	3mix2d 1338	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
3		tglngval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
4		tglngval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		tglngval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
6		tglngval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		tglngval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		tglngval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9		tgcolg.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
10	3, 4, 5, 6, 7, 8, 9	tgcolg 28499	. 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
11	2, 10	mpbird 257	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 847 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 TarskiGcstrkg 28372 Itvcitv 28378 LineGclng 28379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6438 df-fun 6484 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-trkgc 28393 df-trkgcb 28395 df-trkg 28398

This theorem is referenced by: tgdim01ln 28509 lnxfr 28511 tgbtwnconn1lem3 28519 tgbtwnconnln1 28525 tgbtwnconnln2 28526 tglineeltr 28576

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