Theorem isnrm2 23496

Description: An alternate characterization of normality. This is the important property in the proof of Urysohn's lemma. (Contributed by Jeff Hankins, 1-Feb-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
isnrm2	⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))

Distinct variable group:   𝑐,𝑑,𝑜,𝐽

Proof of Theorem isnrm2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nrmtop 23474	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep2 23494	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
3	2	3exp2 1373	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
4	3	impd 415	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
5	4	ralrimivv 3206	. . 3 ⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
6	1, 5	jca 520	. 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7		simpl 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	opncld 23171	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
10	9	adantr 485	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
11		ineq2 4168	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (𝑐 ∩ 𝑑) = (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)))
12	11	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
13		ineq2 4168	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
15	14	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)))
16	15	rexbidv 3189	. . . . . . . . . . 11 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)))
17	12, 16	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) ↔ ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))))
18	17	rspcv 3578	. . . . . . . . 9 ⊢ ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))))
19	10, 18	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))))
20		inssdif0 4330	. . . . . . . . . 10 ⊢ ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
21	8	cldss 23167	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ ∪ 𝐽)
22	21	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 ⊆ ∪ 𝐽)
23		dfss2 3924	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝐽 ↔ (𝑐 ∩ ∪ 𝐽) = 𝑐)
24	22, 23	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 ∩ ∪ 𝐽) = 𝑐)
25	24	sseq1d 3969	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ 𝑐 ⊆ 𝑥))
26	20, 25	bitr3id 288	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ 𝑐 ⊆ 𝑥))
27		inssdif0 4330	. . . . . . . . . . . 12 ⊢ ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
28		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29		elssuni 4905	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
30	8	clsss3 23197	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
31	28, 29, 30	syl2an 607	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
32		dfss2 3924	. . . . . . . . . . . . . 14 ⊢ (((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
33	31, 32	sylib 221	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
34	33	sseq1d 3969	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
35	27, 34	bitr3id 288	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
36	35	anbi2d 641	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅) ↔ (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
37	36	rexbidva 3187	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅) ↔ ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
38	26, 37	imbi12d 347	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)) ↔ (𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
39	19, 38	sylibd 242	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → (𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
40	39	ralimdva 3177	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑐 ∈ (Clsd‘𝐽)(𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
41		elin 3922	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42		velpw 4568	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 𝑥 ↔ 𝑐 ⊆ 𝑥)
43	42	anbi2i 634	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
44	41, 43	bitri 278	. . . . . . . . 9 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
45	44	imbi1i 352	. . . . . . . 8 ⊢ ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
46		impexp 455	. . . . . . . 8 ⊢ (((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
47	45, 46	bitri 278	. . . . . . 7 ⊢ ((𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)) ↔ (𝑐 ∈ (Clsd‘𝐽) → (𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))))
48	47	ralbii2 3107	. . . . . 6 ⊢ (∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥) ↔ ∀𝑐 ∈ (Clsd‘𝐽)(𝑐 ⊆ 𝑥 → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
49	40, 48	imbitrrdi 255	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
50	49	ralrimdva 3165	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)) → ∀𝑥 ∈ 𝐽 ∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
51	50	imp 411	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → ∀𝑥 ∈ 𝐽 ∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
52		isnrm 23473	. . 3 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ ((cls‘𝐽)‘𝑜) ⊆ 𝑥)))
53	7, 51, 52	sylanbrc 594	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Nrm)
54	6, 53	impbii 212	1 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  ∪ cuni 4873  ‘cfv 6538  Topctop 23031  Clsdccld 23154  clsccl 23156  Nrmcnrm 23448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23032  df-cld 23157  df-cls 23159  df-nrm 23455

This theorem is referenced by:  isnrm3  23497