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| 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 28611 | . . 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 1143 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Itvcitv 28442 hlGchlg 28609 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-hlg 28610 | 
| This theorem is referenced by: hltr 28619 hlperpnel 28734 opphllem4 28759 opphllem5 28760 opphl 28763 hlpasch 28765 colhp 28779 iscgra1 28819 cgrane3 28823 cgrane4 28824 cgracgr 28827 inaghl 28854 | 
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