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Mirrors > Home > MPE Home > Th. List > lejoin2 | Structured version Visualization version GIF version |
Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.) |
Ref | Expression |
---|---|
joinval2.b | ⊢ 𝐵 = (Base‘𝐾) |
joinval2.l | ⊢ ≤ = (le‘𝐾) |
joinval2.j | ⊢ ∨ = (join‘𝐾) |
joinval2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joinval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
joinval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
joinlem.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
Ref | Expression |
---|---|
lejoin2 | ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinval2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | joinval2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | joinval2.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | joinval2.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | joinval2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | joinval2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | joinlem.e | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | joinlem 17450 | . 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))) |
9 | 8 | simplrd 766 | 1 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 〈cop 4478 class class class wbr 4962 dom cdm 5443 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 lecple 16401 joincjn 17383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-lub 17413 df-join 17415 |
This theorem is referenced by: joinle 17453 latlej2 17500 |
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