Theorem iscnrm3llem2 49580

Description: Lemma for iscnrm3l 49581. 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 45634.) (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 3965	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝐶 ⊆ 𝑙 ↔ 𝐶 ⊆ (𝑛 ∩ 𝑍)))
2		ineq1 4168	. . . 4 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → (𝑙 ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ 𝑘))
3	2	eqeq1d 2767	. . 3 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝑙 ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅))
4	1, 3	3anbi13d 1462	. 2 ⊢ (𝑙 = (𝑛 ∩ 𝑍) → ((𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅)))
5		sseq2 3965	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (𝐷 ⊆ 𝑘 ↔ 𝐷 ⊆ (𝑚 ∩ 𝑍)))
6		ineq2 4169	. . . 4 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝑛 ∩ 𝑍) ∩ 𝑘) = ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)))
7	6	eqeq1d 2767	. . 3 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → (((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅))
8	5, 7	3anbi23d 1463	. 2 ⊢ (𝑘 = (𝑚 ∩ 𝑍) → ((𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ 𝑘 ∧ ((𝑛 ∩ 𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛 ∩ 𝑍) ∧ 𝐷 ⊆ (𝑚 ∩ 𝑍) ∧ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)))
9		simp11 1220	. . . 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 1322	. . . 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 1216	. . . 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 17469	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑛 ∈ 𝐽) → (𝑛 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
13	9, 10, 11, 12	syl3anc 1394	. . 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 1217	. . . 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 17469	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝑚 ∈ 𝐽) → (𝑚 ∩ 𝑍) ∈ (𝐽 ↾t 𝑍))
16	9, 10, 14, 15	syl3anc 1394	. . 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 1226	. . . . 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 2766	. . . . . 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 4567	. . . . . 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 2766	. . . . . 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 1323	. . . . . 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 49543	. . . . 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 4195	. . . 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 1227	. . . . 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 1324	. . . . . 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 49543	. . . . 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 4195	. . . 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 4191	. . . . . . 7 ⊢ (𝑛 ∩ 𝑍) ⊆ 𝑛
29		inss1 4191	. . . . . . 7 ⊢ (𝑚 ∩ 𝑍) ⊆ 𝑚
30		ss2in 4199	. . . . . . 7 ⊢ (((𝑛 ∩ 𝑍) ⊆ 𝑛 ∧ (𝑚 ∩ 𝑍) ⊆ 𝑚) → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚))
31	28, 29, 30	mp2an 704	. . . . . 6 ⊢ ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ (𝑛 ∩ 𝑚)
32		simp33 1228	. . . . . 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 3981	. . . . 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 4359	. . . . 5 ⊢ (((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) ⊆ ∅ → ((𝑛 ∩ 𝑍) ∩ (𝑚 ∩ 𝑍)) = ∅)
35	33, 34	syl 18	. . . 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 1144	. . 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 1144	. 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 49569	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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  ∪ cuni 4867  ‘cfv 6525  (class class class)co 7400   ↾_t crest 17461  Topctop 23007  Clsdccld 23130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-en 8932  df-fin 8935  df-fi 9359  df-rest 17463  df-topgen 17484  df-top 23008  df-topon 23025  df-bases 23060  df-cld 23133

This theorem is referenced by:  iscnrm3l  49581