Theorem cdleme0nex 37586

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 37507- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 36639, 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 1199	. . . 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 1201	. . . 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 515	. . 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 1198	. . . . . 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 1202	. . . . . . 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 3199	. . . . . . 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 5033	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊))
9	8	notbid 321	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑅 ≤ 𝑊))
10		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑅))
11		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	10, 11	eqeq12d 2814	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
13	9, 12	anbi12d 633	. . . . . . . 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 3569	. . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ ∀𝑟 ∈ 𝐴 ¬ (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))) → ¬ (¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
16	4, 7, 15	syl2anc 587	. . . . 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 1200	. . . . . . . 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 36655	. . . . . . . 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 1203	. . . . . . 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 1204	. . . . . . 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 1205	. . . . . . 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 36639	. . . . . . 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 1380	. . . . . 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 631	. . . . 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 979	. . . . 5 ⊢ (¬ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∨ ¬ (¬ 𝑅 ≤ 𝑊 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
31		df-3an 1086	. . . . . . . 8 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
32	31	anbi2i 625	. . . . . . 7 ⊢ ((¬ 𝑅 ≤ 𝑊 ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) ↔ (¬ 𝑅 ≤ 𝑊 ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
33		an12 644	. . . . . . 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		pm4.62 853	. . . . 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 138	. 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 3079	. . 3 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ↔ ¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
41	40	con2bii 361	. 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 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  lecple 16564  joincjn 17546  Atomscatm 36559  CvLatclc 36561  HLchlt 36646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647

This theorem is referenced by:  cdleme18c  37589  cdleme18d  37591  cdlemg17b  37958  cdlemg17h  37964