Theorem 4atex 40700

Description: Whenever there are at least 4 atoms under 𝑃 ∨ 𝑄 (specifically, 𝑃, 𝑄, 𝑟, and (𝑃 ∨ 𝑄) ∧ 𝑊), there are also at least 4 atoms under 𝑃 ∨ 𝑆. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p ∨ q/0 and hence p ∨ s/0 contains at least four atoms..." Note that by cvlsupr2 39967, our (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) is a shorter way to express 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄). (Contributed by NM, 27-May-2013.)

Hypotheses

Ref	Expression
4that.l	⊢ ≤ = (le‘𝐾)
4that.j	⊢ ∨ = (join‘𝐾)
4that.a	⊢ 𝐴 = (Atoms‘𝐾)
4that.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
4atex	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Distinct variable groups:   𝑧,𝑟,𝐴   𝐻,𝑟   ∨ ,𝑟,𝑧   𝐾,𝑟,𝑧   ≤ ,𝑟,𝑧   𝑃,𝑟,𝑧   𝑄,𝑟,𝑧   𝑆,𝑟,𝑧   𝑊,𝑟,𝑧

Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem 4atex

Step	Hyp	Ref	Expression
1		simp21l 1304	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ∈ 𝐴)
2	1	ad2antrr 736	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) → 𝑃 ∈ 𝐴)
3		simp21r 1305	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ¬ 𝑃 ≤ 𝑊)
4	3	ad2antrr 736	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) → ¬ 𝑃 ≤ 𝑊)
5		oveq1 7403	. . . . . 6 ⊢ (𝑃 = 𝑆 → (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
6	5	eqcoms 2770	. . . . 5 ⊢ (𝑆 = 𝑃 → (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
7	6	adantl 485	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) → (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
8		breq1 5103	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑧 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
9	8	notbid 320	. . . . . 6 ⊢ (𝑧 = 𝑃 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
10		oveq2 7404	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑃))
11		oveq2 7404	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑃))
12	10, 11	eqeq12d 2778	. . . . . 6 ⊢ (𝑧 = 𝑃 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃)))
13	9, 12	anbi12d 641	. . . . 5 ⊢ (𝑧 = 𝑃 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑃 ≤ 𝑊 ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))))
14	13	rspcev 3581	. . . 4 ⊢ ((𝑃 ∈ 𝐴 ∧ (¬ 𝑃 ≤ 𝑊 ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
15	2, 4, 7, 14	syl12anc 847	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 = 𝑃) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
16		simpl3r 1243	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
17	16	ad2antrr 736	. . . . 5 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 = 𝑄) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
18		breq1 5103	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑟 ≤ 𝑊 ↔ 𝑧 ≤ 𝑊))
19	18	notbid 320	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑧 ≤ 𝑊))
20		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑧))
21		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑧))
22	20, 21	eqeq12d 2778	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
23	19, 22	anbi12d 641	. . . . . . . 8 ⊢ (𝑟 = 𝑧 → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
24	23	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
25		oveq1 7403	. . . . . . . . . 10 ⊢ (𝑆 = 𝑄 → (𝑆 ∨ 𝑧) = (𝑄 ∨ 𝑧))
26	25	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝑆 = 𝑄 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
27	26	anbi2d 639	. . . . . . . 8 ⊢ (𝑆 = 𝑄 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
28	27	rexbidv 3186	. . . . . . 7 ⊢ (𝑆 = 𝑄 → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
29	24, 28	bitr4id 292	. . . . . 6 ⊢ (𝑆 = 𝑄 → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
30	29	adantl 485	. . . . 5 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 = 𝑄) → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
31	17, 30	mpbid 234	. . . 4 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 = 𝑄) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
32		simp22l 1306	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
33	32	ad3antrrr 740	. . . . 5 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑄 ∈ 𝐴)
34		simp22r 1307	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ¬ 𝑄 ≤ 𝑊)
35	34	ad3antrrr 740	. . . . 5 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → ¬ 𝑄 ≤ 𝑊)
36		simp3l 1215	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ≠ 𝑄)
37	36	necomd 3012	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑄 ≠ 𝑃)
38	37	ad3antrrr 740	. . . . . 6 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑄 ≠ 𝑃)
39		simpr 488	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑆 ≠ 𝑄)
40	39	necomd 3012	. . . . . 6 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑄 ≠ 𝑆)
41		simpllr 785	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑆 ≤ (𝑃 ∨ 𝑄))
42		simp1l 1211	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐾 ∈ HL)
43		hlcvl 39983	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
44	42, 43	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐾 ∈ CvLat)
45	44	ad3antrrr 740	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝐾 ∈ CvLat)
46		simp23 1222	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑆 ∈ 𝐴)
47	46	ad3antrrr 740	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑆 ∈ 𝐴)
48	1	ad3antrrr 740	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑃 ∈ 𝐴)
49		simplr 778	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑆 ≠ 𝑃)
50		4that.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
51		4that.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
52		4that.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
53	50, 51, 52	cvlatexch1 39960	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑆 ≠ 𝑃) → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆)))
54	45, 47, 33, 48, 49, 53	syl131anc 1402	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆)))
55	41, 54	mpd 15	. . . . . 6 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆))
56	49	necomd 3012	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → 𝑃 ≠ 𝑆)
57	52, 50, 51	cvlsupr2 39967	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑆) → ((𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄) ↔ (𝑄 ≠ 𝑃 ∧ 𝑄 ≠ 𝑆 ∧ 𝑄 ≤ (𝑃 ∨ 𝑆))))
58	45, 48, 47, 33, 56, 57	syl131anc 1402	. . . . . 6 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → ((𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄) ↔ (𝑄 ≠ 𝑃 ∧ 𝑄 ≠ 𝑆 ∧ 𝑄 ≤ (𝑃 ∨ 𝑆))))
59	38, 40, 55, 58	mpbir3and 1356	. . . . 5 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))
60		breq1 5103	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑧 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
61	60	notbid 320	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
62		oveq2 7404	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑄))
63		oveq2 7404	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑄))
64	62, 63	eqeq12d 2778	. . . . . . 7 ⊢ (𝑧 = 𝑄 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄)))
65	61, 64	anbi12d 641	. . . . . 6 ⊢ (𝑧 = 𝑄 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))))
66	65	rspcev 3581	. . . . 5 ⊢ ((𝑄 ∈ 𝐴 ∧ (¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
67	33, 35, 59, 66	syl12anc 847	. . . 4 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) ∧ 𝑆 ≠ 𝑄) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
68	31, 67	pm2.61dane 3044	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑆 ≠ 𝑃) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
69	15, 68	pm2.61dane 3044	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
70		simpl1 1205	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
71		simpl2 1206	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴))
72		simpl3l 1242	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄)
73		simpr 488	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
74		simpl3r 1243	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
75		4that.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
76	50, 51, 52, 75	4atexlem7 40699	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
77	70, 71, 72, 73, 74, 76	syl113anc 1401	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
78	69, 77	pm2.61dan 822	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  lecple 17293  joincjn 18343  Atomscatm 39887  CvLatclc 39889  HLchlt 39974  LHypclh 40608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p0 18455  df-p1 18456  df-lat 18464  df-clat 18531  df-oposet 39800  df-ol 39802  df-oml 39803  df-covers 39890  df-ats 39891  df-atl 39922  df-cvlat 39946  df-hlat 39975  df-llines 40122  df-lplanes 40123  df-lhyp 40612

This theorem is referenced by:  4atex2  40701