Theorem cdleme0nex 40399

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 40320- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 39452, 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 1203	. . . 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 1205	. . . 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 511	. . 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 1202	. . . . . 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 1206	. . . . . . 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 3058	. . . . . . 7 ⊢ (∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
7	5, 6	sylibr 234	. . . . . 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 5092	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
9	8	notbid 318	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
10		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑅))
11		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	10, 11	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
13	9, 12	anbi12d 632	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
14	13	notbid 318	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ↔ ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))))
15	14	rspcva 3570	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ ∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
16	4, 7, 15	syl2anc 584	. . . . 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 1204	. . . . . . . 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 39468	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
19	17, 18	syl 17	. . . . . . 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 1207	. . . . . . 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 1208	. . . . . . 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 1209	. . . . . . 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 39452	. . . . . . 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 1385	. . . . . 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 630	. . . . 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 324	. . . 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 983	. . . . 5 ⊢ (¬ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∨ ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
31		df-3an 1088	. . . . . . . 8 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
32	31	anbi2i 623	. . . . . . 7 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ 𝑅 ≤ 𝑊 ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
33		an12 645	. . . . . . 7 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
34	32, 33	bitri 275	. . . . . 6 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
35	34	notbii 320	. . . . 5 ⊢ (¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ¬ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
36		pm4.62 856	. . . . 5 ⊢ (((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∨ ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
37	30, 35, 36	3bitr4ri 304	. . . 4 ⊢ (((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
38	29, 37	sylibr 234	. . 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 136	. 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 3021	. . 3 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ↔ ¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
41	40	con2bii 357	. 2 ⊢ ((𝑅 = 𝑃 ∨ 𝑅 = 𝑄) ↔ ¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄))
42	39, 41	sylibr 234	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  lecple 17168  joincjn 18217  Atomscatm 39372  CvLatclc 39374  HLchlt 39459

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 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  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 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  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-lat 18338  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460

This theorem is referenced by:  cdleme18c  40402  cdleme18d  40404  cdlemg17b  40771  cdlemg17h  40777