Theorem iscnrm3llem2 48944

Description: Lemma for iscnrm3l 48945. 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 45043.) (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 3962	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝐶 ⊆ 𝑙 ↔ 𝐶 ⊆ (𝑛 ∩ 𝑍)))
2		ineq1 4164	. . . 4 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝑙 ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ 𝑘))
3	2	eqeq1d 2731	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝑙 ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅))
4	1, 3	3anbi13d 1440	. 2 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅)))
5		sseq2 3962	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (𝐷 ⊆ 𝑘 ↔ 𝐷 ⊆ (𝑚 ∩ 𝑍)))
6		ineq2 4165	. . . 4 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝑛 ∩ 𝑍) ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)))
7	6	eqeq1d 2731	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅))
8	5, 7	3anbi23d 1441	. 2 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ (𝑚 ∩ 𝑍) ∧ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)))
9		simp11 1204	. . . 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 1306	. . . 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 1200	. . . 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 17332	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑛 ∈ 𝐽) → (𝑛 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
13	9, 10, 11, 12	syl3anc 1373	. . 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 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) = ∅)) → 𝑚 ∈ 𝐽)
15		elrestr 17332	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑚 ∈ 𝐽) → (𝑚 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
16	9, 10, 14, 15	syl3anc 1373	. . 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 1210	. . . . 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 2730	. . . . . 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 4560	. . . . . 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 2730	. . . . . 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 1307	. . . . . 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 48907	. . . . 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 4192	. . . 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 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) = ∅)) → 𝐷 ⊆ 𝑚)
25		simp123 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) = ∅)) → 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍)))
26	9, 18, 19, 20, 25	restcls2lem 48907	. . . . 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 4192	. . . 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 4188	. . . . . . 7 ⊢ (𝑛 ∩ 𝑍) ⊆ 𝑛
29		inss1 4188	. . . . . . 7 ⊢ (𝑚 ∩ 𝑍) ⊆ 𝑚
30		ss2in 4196	. . . . . . 7 ⊢ (((𝑛 ∩ 𝑍) ⊆ 𝑛 ∧ (𝑚 ∩ 𝑍) ⊆ 𝑚) → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚))
31	28, 29, 30	mp2an 692	. . . . . 6 ⊢ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚)
32		simp33 1212	. . . . . 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 3978	. . . . 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 4353	. . . . 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 1128	. . 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 1128	. 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 48933	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 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  ‘cfv 6482  (class class class)co 7349   ↾_t crest 17324  Topctop 22778  Clsdccld 22901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  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-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-en 8873  df-fin 8876  df-fi 9301  df-rest 17326  df-topgen 17347  df-top 22779  df-topon 22796  df-bases 22831  df-cld 22904

This theorem is referenced by:  iscnrm3l  48945