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Mirrors > Home > MPE Home > Th. List > ncolne1 | Structured version Visualization version GIF version |
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) |
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 |
---|---|
ncolne1 | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncolne.2 | . . 3 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | |
2 | tglineelsb2.p | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | tglineelsb2.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglineelsb2.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
7 | ncolne.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 7 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐵) |
9 | ncolne.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | 9 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝐵) |
11 | ncolne.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 11 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐵) |
13 | eqid 2778 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
14 | 2, 13, 4, 6, 12, 10 | tgbtwntriv1 25842 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍)) |
15 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
16 | 15 | oveq1d 6937 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍)) |
17 | 14, 16 | eleqtrd 2861 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍)) |
18 | 2, 3, 4, 6, 8, 10, 12, 17 | btwncolg1 25906 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
19 | 1, 18 | mtand 806 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
20 | 19 | neqned 2976 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 |
This theorem is referenced by: ncolne2 25977 tglineneq 25995 midexlem 26043 mideulem2 26082 outpasch 26103 hlpasch 26104 trgcopy 26152 trgcopyeulem 26153 acopy 26182 acopyeu 26183 cgrg3col4 26202 tgasa1 26207 isoas 26213 |
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