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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 26671 could be simplified out and deleted, replaced by ncolcom 26606. |
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 26606 | . 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | ncolne1 26670 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 TarskiGcstrkg 26475 Itvcitv 26481 LineGclng 26482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkg 26498 |
This theorem is referenced by: midexlem 26737 |
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