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Mirrors > Home > MPE Home > Th. List > hleqnid | Structured version Visualization version GIF version |
Description: The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-Mar-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) |
hlln.1 | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
Ref | Expression |
---|---|
hleqnid | ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neirr 2950 | . . 3 ⊢ ¬ 𝐴 ≠ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ¬ 𝐴 ≠ 𝐴) |
3 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
6 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ∈ 𝑃) |
8 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐵 ∈ 𝑃) |
10 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐺 ∈ TarskiG) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴(𝐾‘𝐴)𝐵) | |
13 | 3, 4, 5, 7, 9, 7, 11, 12 | hlne1 27376 | . 2 ⊢ ((𝜑 ∧ 𝐴(𝐾‘𝐴)𝐵) → 𝐴 ≠ 𝐴) |
14 | 2, 13 | mtand 814 | 1 ⊢ (𝜑 → ¬ 𝐴(𝐾‘𝐴)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5103 ‘cfv 6493 Basecbs 17043 TarskiGcstrkg 27198 Itvcitv 27204 hlGchlg 27371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-hlg 27372 |
This theorem is referenced by: mirbtwnhl 27451 |
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