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Mirrors > Home > MPE Home > Th. List > hlne2 | Structured version Visualization version GIF version |
Description: The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ishlg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
hlcomd.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
Ref | Expression |
---|---|
hlne2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcomd.1 | . . 3 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
2 | ishlg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ishlg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ishlg.k | . . . 4 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | ishlg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
9 | 2, 3, 4, 5, 6, 7, 8 | ishlg 28625 | . . 3 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
10 | 1, 9 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
11 | 10 | simp2d 1142 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Itvcitv 28456 hlGchlg 28623 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-hlg 28624 |
This theorem is referenced by: hltr 28633 hlperpnel 28748 opphllem4 28773 opphllem5 28774 opphl 28777 hlpasch 28779 colhp 28793 iscgra1 28833 cgrane3 28837 cgrane4 28838 cgracgr 28841 inaghl 28868 |
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