Theorem 2atm 39632

Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)

Hypotheses

Ref	Expression
2atm.l	⊢ ≤ = (le‘𝐾)
2atm.j	⊢ ∨ = (join‘𝐾)
2atm.m	⊢ ∧ = (meet‘𝐾)
2atm.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
2atm	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)))

Proof of Theorem 2atm

Step	Hyp	Ref	Expression
1		simp31 1210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 ≤ (𝑃 ∨ 𝑄))
2		simp32 1211	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 ≤ (𝑅 ∨ 𝑆))
3		simp11 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝐾 ∈ HL)
4	3	hllatd 39469	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝐾 ∈ Lat)
5		simp23 1209	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 ∈ 𝐴)
6		eqid 2731	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		2atm.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 39394	. . . . 5 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
9	5, 8	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 ∈ (Base‘𝐾))
10		simp12 1205	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑃 ∈ 𝐴)
11	6, 7	atbase 39394	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
12	10, 11	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑃 ∈ (Base‘𝐾))
13		simp13 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑄 ∈ 𝐴)
14	6, 7	atbase 39394	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
15	13, 14	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑄 ∈ (Base‘𝐾))
16		2atm.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
17	6, 16	latjcl 18351	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
18	4, 12, 15, 17	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
19		simp21 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑅 ∈ 𝐴)
20		simp22 1208	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑆 ∈ 𝐴)
21	6, 16, 7	hlatjcl 39472	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
22	3, 19, 20, 21	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
23		2atm.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
24		2atm.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
25	6, 23, 24	latlem12 18378	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆)) ↔ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆))))
26	4, 9, 18, 22, 25	syl13anc 1374	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → ((𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆)) ↔ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆))))
27	1, 2, 26	mpbi2and 712	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)))
28		hlatl 39465	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
29	3, 28	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝐾 ∈ AtLat)
30	6, 24	latmcl 18352	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ (Base‘𝐾))
31	4, 18, 22, 30	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ (Base‘𝐾))
32		eqid 2731	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	6, 23, 32, 7	atlen0 39415	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ 𝑇 ∈ 𝐴) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ≠ (0.‘𝐾))
34	29, 31, 5, 27, 33	syl31anc 1375	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ≠ (0.‘𝐾))
35	34	neneqd 2933	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → ¬ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = (0.‘𝐾))
36		simp33 1212	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))
37	16, 24, 32, 7	2atmat0 39631	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = (0.‘𝐾)))
38	3, 10, 13, 19, 20, 36, 37	syl33anc 1387	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = (0.‘𝐾)))
39	38	ord 864	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (¬ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴 → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) = (0.‘𝐾)))
40	35, 39	mt3d 148	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴)
41	23, 7	atcmp 39416	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑇 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ∈ 𝐴) → (𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ↔ 𝑇 = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆))))
42	29, 5, 40, 41	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → (𝑇 ≤ ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)) ↔ 𝑇 = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆))))
43	27, 42	mpbid 232	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑅 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄) ≠ (𝑅 ∨ 𝑆))) → 𝑇 = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  Basecbs 17126  lecple 17174  joincjn 18223  meetcmee 18224  0.cp0 18333  Latclat 18343  Atomscatm 39368  AtLatcal 39369  HLchlt 39455

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 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-proset 18206  df-poset 18225  df-plt 18240  df-lub 18256  df-glb 18257  df-join 18258  df-meet 18259  df-p0 18335  df-lat 18344  df-clat 18411  df-oposet 39281  df-ol 39283  df-oml 39284  df-covers 39371  df-ats 39372  df-atl 39403  df-cvlat 39427  df-hlat 39456  df-llines 39603

This theorem is referenced by:  cdlemk12  40955  cdlemk12u  40977  cdlemk47  41054