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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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
7 | ncolne.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐵) |
9 | ncolne.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝐵) |
11 | ncolne.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐵) |
13 | eqid 2610 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
14 | 2, 13, 4, 6, 12, 10 | tgbtwntriv1 25186 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍)) |
15 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
16 | 15 | oveq1d 6564 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍)) |
17 | 14, 16 | eleqtrd 2690 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍)) |
18 | 2, 3, 4, 6, 8, 10, 12, 17 | btwncolg1 25250 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
19 | 1, 18 | mtand 689 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
20 | 19 | neqned 2789 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 |
This theorem is referenced by: ncolne2 25321 tglineneq 25339 midexlem 25387 mideulem2 25426 outpasch 25447 hlpasch 25448 trgcopy 25496 trgcopyeulem 25497 acopy 25524 acopyeu 25525 cgrg3col4 25534 tgasa1 25539 isoas 25544 |
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