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Mirrors > Home > MPE Home > Th. List > ncolne2 | Structured version Visualization version GIF version |
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 26406 could be simplified out and deleted, replaced by ncolcom 26341. |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ncolne.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ncolne.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ncolne.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ncolne.2 | ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
Ref | Expression |
---|---|
ncolne2 | ⊢ (𝜑 → 𝑋 ≠ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tglineelsb2.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | tglineelsb2.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | ncolne.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ncolne.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ncolne.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | ncolne.2 | . . 3 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | |
9 | 1, 3, 2, 4, 7, 6, 5, 8 | ncolcom 26341 | . 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | ncolne1 26405 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 TarskiGcstrkg 26210 Itvcitv 26216 LineGclng 26217 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
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 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: midexlem 26472 |
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