Theorem clsval2 20854

Description: Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsval2	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Proof of Theorem clsval2

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2921	. . . . . 6 ⊢ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)}
2		clscld.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
3	2	cldopn 20835	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑧) ∈ 𝐽)
4	3	ad2antrl 764	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝐽)
5		sscon 3744	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
6	5	ad2antll 765	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
7	2	topopn 20711	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8		difexg 4808	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑧) ∈ V)
9		elpwg 4166	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∖ 𝑧) ∈ V → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
10	7, 8, 9	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
11	10	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
12	6, 11	mpbird 247	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆))
13	4, 12	elind 3798	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
14	2	cldss 20833	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (Clsd‘𝐽) → 𝑧 ⊆ 𝑋)
15	14	ad2antrl 764	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 ⊆ 𝑋)
16		dfss4 3858	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
17	15, 16	sylib 208	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
18	17	eqcomd 2628	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧)))
19		difeq2 3722	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑋 ∖ 𝑧) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑧)))
20	19	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ 𝑧) → (𝑧 = (𝑋 ∖ 𝑥) ↔ 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧))))
21	20	rspcev 3309	. . . . . . . . . 10 ⊢ (((𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥))
22	13, 18, 21	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥))
23	22	ex 450	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
24		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
25		elin 3796	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆)))
26	25	simplbi 476	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝐽)
27	2	opncld 20837	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
28	24, 26, 27	syl2an 494	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
29	25	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
30	29	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
31	30	elpwid 4170	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ (𝑋 ∖ 𝑆))
32	31	difss2d 3740	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ 𝑋)
33		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ 𝑋)
34		ssconb 3743	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑋) → (𝑥 ⊆ (𝑋 ∖ 𝑆) ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
35	32, 33, 34	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑥 ⊆ (𝑋 ∖ 𝑆) ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
36	31, 35	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
37	28, 36	jca 554	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽)))
39		sseq2 3627	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
40	38, 39	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥))))
41	37, 40	syl5ibrcom 237	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
42	41	rexlimdva 3031	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
43	23, 42	impbid 202	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
44	43	abbidv 2741	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
45	1, 44	syl5eq 2668	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
46	45	inteqd 4480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
47		difexg 4808	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑥) ∈ V)
48	47	ralrimivw 2967	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V)
49		dfiin2g 4553	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
50	7, 48, 49	3syl 18	. . . . 5 ⊢ (𝐽 ∈ Top → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
51	50	adantr 481	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
52	46, 51	eqtr4d 2659	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
53	2	clsval 20841	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧})
54		uniiun 4573	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) = ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥
55	54	difeq2i 3725	. . . . 5 ⊢ (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥)
56	55	a1i 11	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
57		0opn 20709	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
58	57	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝐽)
59		0elpw 4834	. . . . . . 7 ⊢ ∅ ∈ 𝒫 (𝑋 ∖ 𝑆)
60	59	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝒫 (𝑋 ∖ 𝑆))
61	58, 60	elind 3798	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
62		ne0i 3921	. . . . 5 ⊢ (∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅)
63		iindif2 4589	. . . . 5 ⊢ ((𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅ → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
64	61, 62, 63	3syl 18	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
65	56, 64	eqtr4d 2659	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
66	52, 53, 65	3eqtr4d 2666	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
67		difssd 3738	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ 𝑆) ⊆ 𝑋)
68	2	ntrval 20840	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
69	67, 68	sylan2 491	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
70	69	difeq2d 3728	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
71	66, 70	eqtr4d 2659	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475  ∪ ciun 4520  ∩ ciin 4521  ‘cfv 5888  Topctop 20698  Clsdccld 20820  intcnt 20821  clsccl 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  ntrval2  20855  clsdif  20857  cmclsopn  20866  bcth3  23128