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Mirrors > Home > MPE Home > Th. List > Mathboxes > lubprdm | Structured version Visualization version GIF version |
Description: The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.) |
Ref | Expression |
---|---|
lubpr.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
lubpr.b | ⊢ 𝐵 = (Base‘𝐾) |
lubpr.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
lubpr.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
lubpr.l | ⊢ ≤ = (le‘𝐾) |
lubpr.c | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
lubpr.s | ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) |
lubpr.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
lubprdm | ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubpr.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
2 | lubpr.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | lubpr.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
4 | lubpr.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | lubpr.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | lubpr.c | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
7 | lubpr.s | . . 3 ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) | |
8 | lubpr.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lubprlem 48758 | . 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌)) |
10 | 9 | simpld 494 | 1 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {cpr 4632 class class class wbr 5147 dom cdm 5688 ‘cfv 6562 Basecbs 17244 lecple 17304 Posetcpo 18364 lubclub 18366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-proset 18351 df-poset 18370 df-lub 18403 |
This theorem is referenced by: glbprlem 48761 toslat 48770 |
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