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Mirrors > Home > MPE Home > Th. List > glbeu | Structured version Visualization version GIF version |
Description: Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
glbval.b | ⊢ 𝐵 = (Base‘𝐾) |
glbval.l | ⊢ ≤ = (le‘𝐾) |
glbval.g | ⊢ 𝐺 = (glb‘𝐾) |
glbval.p | ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
glbva.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
glbval.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) |
Ref | Expression |
---|---|
glbeu | ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | glbval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | |
2 | glbval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | glbval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | glbval.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
5 | glbval.p | . . . 4 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) | |
6 | glbva.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
7 | 2, 3, 4, 5, 6 | glbeldm 17713 | . . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |
8 | 1, 7 | mpbid 235 | . 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
9 | 8 | simprd 499 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃!wreu 3055 ⊆ wss 3841 class class class wbr 5027 dom cdm 5519 ‘cfv 6333 Basecbs 16579 lecple 16668 glbcglb 17662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-glb 17694 |
This theorem is referenced by: glbval 17716 glbcl 17717 glbprop 17718 meeteu 17743 isglbd 17836 |
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