Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hlne1 | 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 |
---|---|
hlne1 | ⊢ (𝜑 → 𝐴 ≠ 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 26382 | . . 3 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
10 | 1, 9 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)))) |
11 | 10 | simp1d 1138 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Itvcitv 26216 hlGchlg 26380 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-hlg 26381 |
This theorem is referenced by: hleqnid 26388 hltr 26390 opphllem4 26530 opphl 26534 cgrane1 26592 cgrane2 26593 cgrahl 26607 sacgr 26611 acopy 26613 acopyeu 26614 inaghl 26625 |
Copyright terms: Public domain | W3C validator |