Theorem 4atex 40078

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 39344, 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 1291	. . . . 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 726	. . . 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 1292	. . . . 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 726	. . . 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 7438	. . . . . 6 ⊢ (𝑃 = 𝑆 → (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
6	5	eqcoms 2745	. . . . 5 ⊢ (𝑆 = 𝑃 → (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))
7	6	adantl 481	. . . 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 5146	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑧 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
9	8	notbid 318	. . . . . 6 ⊢ (𝑧 = 𝑃 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
10		oveq2 7439	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑃))
11		oveq2 7439	. . . . . . 7 ⊢ (𝑧 = 𝑃 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑃))
12	10, 11	eqeq12d 2753	. . . . . 6 ⊢ (𝑧 = 𝑃 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃)))
13	9, 12	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝑃 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑃 ≤ 𝑊 ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))))
14	13	rspcev 3622	. . . 4 ⊢ ((𝑃 ∈ 𝐴 ∧ (¬ 𝑃 ≤ 𝑊 ∧ (𝑃 ∨ 𝑃) = (𝑆 ∨ 𝑃))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
15	2, 4, 7, 14	syl12anc 837	. . 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 1230	. . . . . 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 726	. . . . 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 5146	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑟 ≤ 𝑊 ↔ 𝑧 ≤ 𝑊))
19	18	notbid 318	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑧 ≤ 𝑊))
20		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑧))
21		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑧))
22	20, 21	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
23	19, 22	anbi12d 632	. . . . . . . 8 ⊢ (𝑟 = 𝑧 → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
24	23	cbvrexvw 3238	. . . . . . 7 ⊢ (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
25		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑆 = 𝑄 → (𝑆 ∨ 𝑧) = (𝑄 ∨ 𝑧))
26	25	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑆 = 𝑄 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧)))
27	26	anbi2d 630	. . . . . . . 8 ⊢ (𝑆 = 𝑄 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
28	27	rexbidv 3179	. . . . . . 7 ⊢ (𝑆 = 𝑄 → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑄 ∨ 𝑧))))
29	24, 28	bitr4id 290	. . . . . 6 ⊢ (𝑆 = 𝑄 → (∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
30	29	adantl 481	. . . . 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 232	. . . 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 1293	. . . . . 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 730	. . . . 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 1294	. . . . . 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 730	. . . . 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 1202	. . . . . . . 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 2996	. . . . . . 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 730	. . . . . 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 484	. . . . . . 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 2996	. . . . . 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 776	. . . . . . 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 1198	. . . . . . . . . 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 39360	. . . . . . . . . 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 730	. . . . . . . 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 1209	. . . . . . . . 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 730	. . . . . . . 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 730	. . . . . . . 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 769	. . . . . . . 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 39337	. . . . . . . 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 1385	. . . . . . 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 2996	. . . . . . 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 39344	. . . . . . 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 1385	. . . . . 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 1343	. . . . 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 5146	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑧 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
61	60	notbid 318	. . . . . . 7 ⊢ (𝑧 = 𝑄 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
62		oveq2 7439	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝑄))
63		oveq2 7439	. . . . . . . 8 ⊢ (𝑧 = 𝑄 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝑄))
64	62, 63	eqeq12d 2753	. . . . . . 7 ⊢ (𝑧 = 𝑄 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄)))
65	61, 64	anbi12d 632	. . . . . 6 ⊢ (𝑧 = 𝑄 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))))
66	65	rspcev 3622	. . . . 5 ⊢ ((𝑄 ∈ 𝐴 ∧ (¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑆 ∨ 𝑄))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
67	33, 35, 59, 66	syl12anc 837	. . . 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 3029	. . 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 3029	. 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 1192	. . 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 1193	. . 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 1229	. . 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 484	. . 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 1230	. . 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 40077	. . 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 1384	. 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 813	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  CvLatclc 39266  HLchlt 39351  LHypclh 39986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lhyp 39990

This theorem is referenced by:  4atex2  40079