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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 26977 could be simplified out and deleted, replaced by ncolcom 26912. |
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 26912 | . 2 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌)) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | ncolne1 26976 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 TarskiGcstrkg 26778 Itvcitv 26784 LineGclng 26785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6389 df-fun 6433 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-trkgc 26799 df-trkgb 26800 df-trkgcb 26801 df-trkg 26804 |
This theorem is referenced by: midexlem 27043 |
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