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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemkl | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 37371. (Contributed by NM, 23-Nov-2012.) |
Ref | Expression |
---|---|
4thatlem.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
Ref | Expression |
---|---|
4atexlemkl | ⊢ (𝜑 → 𝐾 ∈ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4thatlem.ph | . . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | |
2 | 1 | 4atexlemk 37343 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | hllatd 36660 | 1 ⊢ (𝜑 → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 Latclat 17647 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-atl 36594 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: 4atexlemunv 37362 4atexlemtlw 37363 4atexlemc 37365 4atexlemnclw 37366 4atexlemex2 37367 4atexlemcnd 37368 |
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