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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 16994 | . . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |
8 | 1, 7 | mpbid 222 | . 2 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
9 | 8 | simprd 479 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 Basecbs 15857 lecple 15948 glbcglb 16943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-glb 16975 |
This theorem is referenced by: glbval 16997 glbcl 16998 glbprop 16999 meeteu 17024 isglbd 17117 |
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