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Mirrors > Home > MPE Home > Th. List > lubeu | Structured version Visualization version GIF version |
Description: Unique existence proper of a member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
lubval.b | ⊢ 𝐵 = (Base‘𝐾) |
lubval.l | ⊢ ≤ = (le‘𝐾) |
lubval.u | ⊢ 𝑈 = (lub‘𝐾) |
lubval.p | ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
lubval.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
lubeleu.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
Ref | Expression |
---|---|
lubeu | ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubeleu.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) | |
2 | lubval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | lubval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | lubval.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
5 | lubval.p | . . . 4 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) | |
6 | lubval.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
7 | 2, 3, 4, 5, 6 | lubeldm 17707 | . . 3 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (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 2114 ∀wral 3053 ∃!wreu 3055 ⊆ wss 3843 class class class wbr 5030 dom cdm 5525 ‘cfv 6339 Basecbs 16586 lecple 16675 lubclub 17668 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
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 2075 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 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-lub 17700 |
This theorem is referenced by: lubval 17710 lubcl 17711 lubprop 17712 joineu 17736 |
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