btwncolg2 - Metamath Proof Explorer

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Theorem btwncolg2 29034

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 1356	. 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 29032	. 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
11	2, 10	mpbird 260	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∨ wo 861 ∨ w3o 1102 = wceq 1569 ∈ wcel 2146 ‘cfv 6547 (class class class)co 7430 Basecbs 17402 TarskiGcstrkg 28904 Itvcitv 28910 LineGclng 28911

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6501 df-fun 6549 df-fv 6555 df-ov 7433 df-oprab 7434 df-mpo 7435 df-trkgc 28925 df-trkgcb 28927 df-trkg 28930

This theorem is used by: tgdim01ln 29042 lnxfr 29044 tgbtwnconn1lem3 29052 tgbtwnconnln1 29058 tgbtwnconnln2 29059 tglineeltr 29114 ragsupplcgra 29360 tgextancd 33083 tgextahpl 33084 tgextamirnhl 33087 tgextahp 33088

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