Theorem 3dim2 39513

Description: Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)

Hypotheses

Ref	Expression
3dim0.j	⊢ ∨ = (join‘𝐾)
3dim0.l	⊢ ≤ = (le‘𝐾)
3dim0.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3dim2	⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))

Distinct variable groups:   𝑠,𝑟,𝐴   ∨ ,𝑟,𝑠   ≤ ,𝑟,𝑠   𝑃,𝑟,𝑠   𝑄,𝑟,𝑠

Allowed substitution hints:   𝐾(𝑠,𝑟)

Proof of Theorem 3dim2

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3dim0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
2		3dim0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		3dim0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 2, 3	3dim1 39512	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ∃𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
5	4	3adant2 1131	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
6		simpl21 1252	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → 𝑢 ∈ 𝐴)
7		simpl22 1253	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → 𝑣 ∈ 𝐴)
8		simp31 1210	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑄 ≠ 𝑢)
9	8	necomd 2983	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑢 ≠ 𝑄)
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → 𝑢 ≠ 𝑄)
11		oveq1 7353	. . . . . . . . . . . . . 14 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
12		simp11 1204	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝐾 ∈ HL)
13		simp13 1206	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑄 ∈ 𝐴)
14	1, 3	hlatjidm 39414	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
15	12, 13, 14	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑄 ∨ 𝑄) = 𝑄)
16	11, 15	sylan9eqr 2788	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (𝑃 ∨ 𝑄) = 𝑄)
17	16	breq2d 5103	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (𝑢 ≤ (𝑃 ∨ 𝑄) ↔ 𝑢 ≤ 𝑄))
18	17	notbid 318	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑢 ≤ 𝑄))
19		hlatl 39405	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
20	12, 19	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝐾 ∈ AtLat)
21		simp21 1207	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑢 ∈ 𝐴)
22	2, 3	atncmp 39357	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ AtLat ∧ 𝑢 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑢 ≤ 𝑄 ↔ 𝑢 ≠ 𝑄))
23	20, 21, 13, 22	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (¬ 𝑢 ≤ 𝑄 ↔ 𝑢 ≠ 𝑄))
24	23	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (¬ 𝑢 ≤ 𝑄 ↔ 𝑢 ≠ 𝑄))
25	18, 24	bitrd 279	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ↔ 𝑢 ≠ 𝑄))
26	10, 25	mpbird 257	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ¬ 𝑢 ≤ (𝑃 ∨ 𝑄))
27		simpl32 1256	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑢))
28	16	oveq1d 7361	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ((𝑃 ∨ 𝑄) ∨ 𝑢) = (𝑄 ∨ 𝑢))
29	28	breq2d 5103	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → (𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ 𝑣 ≤ (𝑄 ∨ 𝑢)))
30	27, 29	mtbird 325	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))
31		breq1 5094	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
32	31	notbid 318	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑢 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))
33		oveq2 7354	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑢 → ((𝑃 ∨ 𝑄) ∨ 𝑟) = ((𝑃 ∨ 𝑄) ∨ 𝑢))
34	33	breq2d 5103	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
35	34	notbid 318	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑢 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
36	32, 35	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑟 = 𝑢 → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
37		breq1 5094	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
38	37	notbid 318	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
39	38	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑠 = 𝑣 → ((¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
40	36, 39	rspc2ev 3590	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
41	6, 7, 26, 30, 40	syl112anc 1376	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
42		simp22 1208	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑣 ∈ 𝐴)
43		simp23 1209	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → 𝑤 ∈ 𝐴)
44	42, 43	jca 511	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
45	44	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
46		simpll1 1213	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
47		simp32 1211	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑢))
48		simp33 1212	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))
49	21, 47, 48	3jca 1128	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑢 ∈ 𝐴 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
50	49	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → (𝑢 ∈ 𝐴 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
51		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → 𝑃 ≠ 𝑄)
52		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → 𝑃 ≤ (𝑄 ∨ 𝑢))
53	1, 2, 3	3dimlem2 39504	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≤ (𝑄 ∨ 𝑢))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
54	46, 50, 51, 52, 53	syl112anc 1376	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
55		3simpc 1150	. . . . . . . . . . 11 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)) → (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
56	54, 55	syl 17	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
57		breq1 5094	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑣 → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑣 ≤ (𝑃 ∨ 𝑄)))
58	57	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑣 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
59		oveq2 7354	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑣 → ((𝑃 ∨ 𝑄) ∨ 𝑟) = ((𝑃 ∨ 𝑄) ∨ 𝑣))
60	59	breq2d 5103	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
61	60	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
62	58, 61	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑣 → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ↔ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))))
63		breq1 5094	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
64	63	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
65	64	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑤 → ((¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)) ↔ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))))
66	62, 65	rspc2ev 3590	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
67	66	3expa 1118	. . . . . . . . . 10 ⊢ (((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
68	45, 56, 67	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑢)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
69	21, 43	jca 511	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
70	69	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
71		simp1 1136	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
72	21, 42	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
73	8, 48	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑄 ≠ 𝑢 ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
74	71, 72, 73	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))))
75	74	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))))
76		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → 𝑃 ≠ 𝑄)
77		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑢))
78		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))
79	1, 2, 3	3dimlem3 39506	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
80	75, 76, 77, 78, 79	syl13anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
81		3simpc 1150	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
83		breq1 5094	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
84	83	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
85	84	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → ((¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
86	36, 85	rspc2ev 3590	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
87	86	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
88	70, 82, 87	syl2anc 584	. . . . . . . . . 10 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
89	72	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
90	8, 47	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢)))
91	71, 72, 90	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢))))
92	91	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢))))
93		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → 𝑃 ≠ 𝑄)
94		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑢))
95		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))
96	1, 2, 3	3dimlem4 39509	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
97	92, 93, 94, 95, 96	syl121anc 1377	. . . . . . . . . . . 12 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
98		3simpc 1150	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
99	97, 98	syl 17	. . . . . . . . . . 11 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
100	40	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
101	89, 99, 100	syl2anc 584	. . . . . . . . . 10 ⊢ ((((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
102	88, 101	pm2.61dan 812	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑢)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
103	68, 102	pm2.61dan 812	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
104	41, 103	pm2.61dane 3015	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))
105	104	3exp 1119	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → ((𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))))
106	105	3expd 1354	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐴 → (𝑤 ∈ 𝐴 → ((𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))))))
107	106	imp32 418	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑤 ∈ 𝐴 → ((𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))))
108	107	rexlimdv 3131	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))))
109	108	rexlimdvva 3189	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (∃𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟))))
110	5, 109	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  lecple 17168  joincjn 18217  Atomscatm 39308  AtLatcal 39309  HLchlt 39395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39221  df-ol 39223  df-oml 39224  df-covers 39311  df-ats 39312  df-atl 39343  df-cvlat 39367  df-hlat 39396

This theorem is referenced by:  3dim3  39514  lhp2lt  40046