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Mirrors > Home > MPE Home > Th. List > btwncolg2 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
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)) |
Ref | Expression |
---|---|
btwncolg2 | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwncolg2.z | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) | |
2 | 1 | 3mix2d 1333 | . 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 26339 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
11 | 2, 10 | mpbird 259 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-trkgc 26233 df-trkgcb 26235 df-trkg 26238 |
This theorem is referenced by: tgdim01ln 26349 lnxfr 26351 tgbtwnconn1lem3 26359 tgbtwnconnln1 26365 tgbtwnconnln2 26366 tglineeltr 26416 |
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