Theorem iscnrm3llem2 49440

Description: Lemma for iscnrm3l 49441. If there exist disjoint open neighborhoods in the original topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 45507.) (Contributed by Zhi Wang, 5-Sep-2024.)

Assertion

Ref	Expression
iscnrm3llem2	⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))

Distinct variable groups:   𝐶,𝑘,𝑙,𝑚,𝑛   𝐷,𝑘,𝑙,𝑚,𝑛   𝑘,𝐽,𝑙,𝑚,𝑛   𝑘,𝑍,𝑙,𝑚,𝑛

Proof of Theorem iscnrm3llem2

Step	Hyp	Ref	Expression
1		sseq2 3949	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝐶 ⊆ 𝑙 ↔ 𝐶 ⊆ (𝑛 ∩ 𝑍)))
2		ineq1 4154	. . . 4 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝑙 ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ 𝑘))
3	2	eqeq1d 2739	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝑙 ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅))
4	1, 3	3anbi13d 1441	. 2 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅)))
5		sseq2 3949	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (𝐷 ⊆ 𝑘 ↔ 𝐷 ⊆ (𝑚 ∩ 𝑍)))
6		ineq2 4155	. . . 4 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝑛 ∩ 𝑍) ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)))
7	6	eqeq1d 2739	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅))
8	5, 7	3anbi23d 1442	. 2 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ (𝑚 ∩ 𝑍) ∧ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)))
9		simp11 1205	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐽 ∈ Top)
10		simp121 1307	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑍 ∈ 𝒫 ∪ 𝐽)
11		simp2l 1201	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ∈ 𝐽)
12		elrestr 17385	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑛 ∈ 𝐽) → (𝑛 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
13	9, 10, 11, 12	syl3anc 1374	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
14		simp2r 1202	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ∈ 𝐽)
15		elrestr 17385	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑚 ∈ 𝐽) → (𝑚 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
16	9, 10, 14, 15	syl3anc 1374	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
17		simp31 1211	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐶 ⊆ 𝑛)
18		eqidd 2738	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ∪ 𝐽 = ∪ 𝐽)
19	10	elpwid 4551	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑍 ⊆ ∪ 𝐽)
20		eqidd 2738	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝐽 ↾t 𝑍) = (𝐽 ↾t 𝑍))
21		simp122 1308	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)))
22	9, 18, 19, 20, 21	restcls2lem 49403	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐶 ⊆ 𝑍)
23	17, 22	ssind 4182	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐶 ⊆ (𝑛 ∩ 𝑍))
24		simp32 1212	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐷 ⊆ 𝑚)
25		simp123 1309	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍)))
26	9, 18, 19, 20, 25	restcls2lem 49403	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐷 ⊆ 𝑍)
27	24, 26	ssind 4182	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐷 ⊆ (𝑚 ∩ 𝑍))
28		inss1 4178	. . . . . . 7 ⊢ (𝑛 ∩ 𝑍) ⊆ 𝑛
29		inss1 4178	. . . . . . 7 ⊢ (𝑚 ∩ 𝑍) ⊆ 𝑚
30		ss2in 4186	. . . . . . 7 ⊢ (((𝑛 ∩ 𝑍) ⊆ 𝑛 ∧ (𝑚 ∩ 𝑍) ⊆ 𝑚) → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚))
31	28, 29, 30	mp2an 693	. . . . . 6 ⊢ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚)
32		simp33 1213	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑚) = ∅)
33	31, 32	sseqtrid 3965	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ ∅)
34		ss0 4343	. . . . 5 ⊢ (((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ ∅ → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)
35	33, 34	syl 17	. . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)
36	23, 27, 35	3jca 1129	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ (𝑚 ∩ 𝑍) ∧ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅))
37	13, 16, 36	3jca 1129	. 2 ⊢ (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑛 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍) ∧ (𝑚 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍) ∧ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ (𝑚 ∩ 𝑍) ∧ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)))
38	4, 8, 37	iscnrm3lem7 49429	1 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6493  (class class class)co 7361   ↾_t crest 17377  Topctop 22871  Clsdccld 22994

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17379  df-topgen 17400  df-top 22872  df-topon 22889  df-bases 22924  df-cld 22997

This theorem is referenced by:  iscnrm3l  49441