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Mirrors > Home > MPE Home > Th. List > lubel | Structured version Visualization version GIF version |
Description: An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
lublem.b | ⊢ 𝐵 = (Base‘𝐾) |
lublem.l | ⊢ ≤ = (le‘𝐾) |
lublem.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
lubel | ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑋 ≤ (𝑈‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatl 18458 | . . . 4 ⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat) | |
2 | ssel 3975 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵)) | |
3 | 2 | impcom 409 | . . . 4 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑋 ∈ 𝐵) |
4 | lublem.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | lublem.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
6 | 4, 5 | lubsn 18432 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
7 | 1, 3, 6 | syl2an 597 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ (𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵)) → (𝑈‘{𝑋}) = 𝑋) |
8 | 7 | 3impb 1116 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
9 | snssi 4811 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → {𝑋} ⊆ 𝑆) | |
10 | lublem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
11 | 4, 10, 5 | lubss 18463 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ {𝑋} ⊆ 𝑆) → (𝑈‘{𝑋}) ≤ (𝑈‘𝑆)) |
12 | 9, 11 | syl3an3 1166 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑈‘{𝑋}) ≤ (𝑈‘𝑆)) |
13 | 12 | 3com23 1127 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → (𝑈‘{𝑋}) ≤ (𝑈‘𝑆)) |
14 | 8, 13 | eqbrtrrd 5172 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑋 ≤ (𝑈‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ‘cfv 6541 Basecbs 17141 lecple 17201 lubclub 18259 Latclat 18381 CLatccla 18448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-proset 18245 df-poset 18263 df-lub 18296 df-glb 18297 df-join 18298 df-meet 18299 df-lat 18382 df-clat 18449 |
This theorem is referenced by: lubun 18465 atlatmstc 38178 2polssN 38775 |
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