Theorem 3dim2 40099

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 40098	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → ∃𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
5	4	3adant2 1147	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ∃𝑢 ∈ 𝐴 ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑄 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑢) ∨ 𝑣)))
6		simpl21 1268	. . . . . . . . 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 1269	. . . . . . . . 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 1226	. . . . . . . . . . . 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 3015	. . . . . . . . . . 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 485	. . . . . . . . . 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 7407	. . . . . . . . . . . . . 14 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
12		simp11 1220	. . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . . 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 40000	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
15	12, 13, 14	syl2anc 595	. . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . 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 5116	. . . . . . . . . . . 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 321	. . . . . . . . . . 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 39991	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
20	12, 19	syl 18	. . . . . . . . . . . . 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 1223	. . . . . . . . . . . . 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 39943	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ AtLat ∧ 𝑢 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑢 ≤ 𝑄 ↔ 𝑢 ≠ 𝑄))
23	20, 21, 13, 22	syl3anc 1394	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 282	. . . . . . . . . 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 260	. . . . . . . . 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 1272	. . . . . . . . . 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 7415	. . . . . . . . . . 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 5116	. . . . . . . . . 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 328	. . . . . . . . 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 5107	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
32	31	notbid 321	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑢 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))
33		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑢 → ((𝑃 ∨ 𝑄) ∨ 𝑟) = ((𝑃 ∨ 𝑄) ∨ 𝑢))
34	33	breq2d 5116	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑢 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
35	34	notbid 321	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑢 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
36	32, 35	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑟 = 𝑢 → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
37		breq1 5107	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
38	37	notbid 321	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
39	38	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑠 = 𝑣 → ((¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
40	36, 39	rspc2ev 3597	. . . . . . . . 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 1397	. . . . . . . 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 1224	. . . . . . . . . . . 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 1225	. . . . . . . . . . . 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 520	. . . . . . . . . . 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 738	. . . . . . . . . 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 1229	. . . . . . . . . . . 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 1227	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . 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 738	. . . . . . . . . . . 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 780	. . . . . . . . . . . 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 489	. . . . . . . . . . . 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 40090	. . . . . . . . . . . 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 1397	. . . . . . . . . . 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 1166	. . . . . . . . . . 11 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)) → (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
56	54, 55	syl 18	. . . . . . . . . 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 5107	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑣 → (𝑟 ≤ (𝑃 ∨ 𝑄) ↔ 𝑣 ≤ (𝑃 ∨ 𝑄)))
58	57	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑣 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑣 ≤ (𝑃 ∨ 𝑄)))
59		oveq2 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑣 → ((𝑃 ∨ 𝑄) ∨ 𝑟) = ((𝑃 ∨ 𝑄) ∨ 𝑣))
60	59	breq2d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
61	60	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
62	58, 61	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑣 → ((¬ 𝑟 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑟)) ↔ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))))
63		breq1 5107	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
64	63	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)))
65	64	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑤 → ((¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣)) ↔ (¬ 𝑣 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑣))))
66	62, 65	rspc2ev 3597	. . . . . . . . . . 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 1134	. . . . . . . . . 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 595	. . . . . . . . 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 520	. . . . . . . . . . . 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 742	. . . . . . . . . . 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 1152	. . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 40092	. . . . . . . . . . . . 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 1395	. . . . . . . . . . . 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 1166	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
82	80, 81	syl 18	. . . . . . . . . . 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 5107	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
84	83	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
85	84	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → ((¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) ↔ (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢))))
86	36, 85	rspc2ev 3597	. . . . . . . . . . . 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 1134	. . . . . . . . . . 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 595	. . . . . . . . . 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 742	. . . . . . . . . . 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 520	. . . . . . . . . . . . . . 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 1144	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 40095	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 1166	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)) → (¬ 𝑢 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑢)))
99	97, 98	syl 18	. . . . . . . . . . 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 1134	. . . . . . . . . . 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 595	. . . . . . . . . 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 824	. . . . . . . . 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 824	. . . . . . . 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 3047	. . . . . . 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 1135	. . . . . 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 1370	. . . . 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 423	. . . 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 3164	. . 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 3222	. 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 16	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089   class class class wbr 5104  ‘cfv 6525  (class class class)co 7400  lecple 17305  joincjn 18355  Atomscatm 39894  AtLatcal 39895  HLchlt 39981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-proset 18338  df-poset 18357  df-plt 18372  df-lub 18388  df-glb 18389  df-join 18390  df-meet 18391  df-p0 18467  df-p1 18468  df-lat 18476  df-clat 18543  df-oposet 39807  df-ol 39809  df-oml 39810  df-covers 39897  df-ats 39898  df-atl 39929  df-cvlat 39953  df-hlat 39982

This theorem is referenced by:  3dim3  40100  lhp2lt  40632