Theorem cdleme0nex 40953

Description: Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p ∨ q/0 (i.e. the sublattice from 0 to p ∨ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 40874- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 40006, our (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) is a shorter way to express 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ (𝑃 ∨ 𝑄). Thus, the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)

Hypotheses

Ref	Expression
cdleme0nex.l	⊢ ≤ = (le‘𝐾)
cdleme0nex.j	⊢ ∨ = (join‘𝐾)
cdleme0nex.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cdleme0nex	⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))

Distinct variable groups:   𝐴,𝑟   ∨ ,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑊,𝑟

Allowed substitution hint:   𝐾(𝑟)

Proof of Theorem cdleme0nex

Step	Hyp	Ref	Expression
1		simp3r 1219	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ¬ 𝑅 ≤ 𝑊)
2		simp12 1221	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑅 ≤ (𝑃 ∨ 𝑄))
3	1, 2	jca 520	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
4		simp3l 1218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑅 ∈ 𝐴)
5		simp13 1222	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
6		ralnex 3097	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
7	5, 6	sylibr 237	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
8		breq1 5116	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
9	8	notbid 321	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
10		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑅))
11		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	10, 11	eqeq12d 2785	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
13	9, 12	anbi12d 643	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
14	13	notbid 321	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
15	14	rspcva 3588	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ ∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
16	4, 7, 15	syl2anc 595	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
17		simp11 1220	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐾 ∈ HL)
18		hlcvl 40022	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
19	17, 18	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐾 ∈ CvLat)
20		simp21 1223	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
21		simp22 1224	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑄 ∈ 𝐴)
22		simp23 1225	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑃 ≠ 𝑄)
23		cdleme0nex.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
24		cdleme0nex.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
25		cdleme0nex.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
26	23, 24, 25	cvlsupr2 40006	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
27	19, 20, 21, 4, 22, 26	syl131anc 1408	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
28	27	anbi2d 641	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ↔ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))))
29	16, 28	mtbid 327	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
30		ianor 997	. . . . 5 ⊢ (¬ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∨ ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
31		df-3an 1103	. . . . . . . 8 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
32	31	anbi2i 634	. . . . . . 7 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ 𝑅 ≤ 𝑊 ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
33		an12 657	. . . . . . 7 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
34	32, 33	bitri 278	. . . . . 6 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
35	34	notbii 323	. . . . 5 ⊢ (¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ¬ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
36		pm4.62 869	. . . . 5 ⊢ (((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∨ ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
37	30, 35, 36	3bitr4ri 307	. . . 4 ⊢ (((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
38	29, 37	sylibr 237	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
39	3, 38	mt2d 137	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄))
40		neanior 3057	. . 3 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ↔ ¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
41	40	con2bii 360	. 2 ⊢ ((𝑅 = 𝑃 ∨ 𝑅 = 𝑄) ↔ ¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄))
42	39, 41	sylibr 237	1 ⊢ (((𝐾 ∈ HL ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  lecple 17316  joincjn 18366  Atomscatm 39926  CvLatclc 39928  HLchlt 40013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-lat 18487  df-covers 39929  df-ats 39930  df-atl 39961  df-cvlat 39985  df-hlat 40014

This theorem is referenced by:  cdleme18c  40956  cdleme18d  40958  cdlemg17b  41325  cdlemg17h  41331