Theorem isnrm2 21162

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 21140	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep2 21160	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
3	2	3exp2 1285	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
4	3	impd 447	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
5	4	ralrimivv 2970	. . 3 ⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
6	1, 5	jca 554	. 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7		simpl 473	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8		eqid 2622	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	opncld 20837	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
10	9	adantr 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
11		ineq2 3808	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (𝑐 ∩ 𝑑) = (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)))
12	11	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
13		ineq2 3808	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
15	14	anbi2d 740	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)))
16	15	rexbidv 3052	. . . . . . . . . . 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 334	. . . . . . . . . 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 3305	. . . . . . . . 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 17	. . . . . . . 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 3947	. . . . . . . . . 10 ⊢ ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
21	8	cldss 20833	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ ∪ 𝐽)
22	21	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 ⊆ ∪ 𝐽)
23		df-ss 3588	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝐽 ↔ (𝑐 ∩ ∪ 𝐽) = 𝑐)
24	22, 23	sylib 208	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 ∩ ∪ 𝐽) = 𝑐)
25	24	sseq1d 3632	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ 𝑐 ⊆ 𝑥))
26	20, 25	syl5bbr 274	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ 𝑐 ⊆ 𝑥))
27		inssdif0 3947	. . . . . . . . . . . 12 ⊢ ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
28		simpll 790	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29		elssuni 4467	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
30	8	clsss3 20863	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
31	28, 29, 30	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
32		df-ss 3588	. . . . . . . . . . . . . 14 ⊢ (((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
33	31, 32	sylib 208	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
34	33	sseq1d 3632	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
35	27, 34	syl5bbr 274	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
36	35	anbi2d 740	. . . . . . . . . 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 3049	. . . . . . . . 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 334	. . . . . . . 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 229	. . . . . . 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 2962	. . . . . 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 3796	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42		selpw 4165	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 𝑥 ↔ 𝑐 ⊆ 𝑥)
43	42	anbi2i 730	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
44	41, 43	bitri 264	. . . . . . . . 9 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
45	44	imbi1i 339	. . . . . . . 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 462	. . . . . . . 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 264	. . . . . . 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 2978	. . . . . 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	syl6ibr 242	. . . . 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 2969	. . . 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 445	. . 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 21139	. . 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 698	. 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 199	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820  clsccl 20822  Nrmcnrm 21114

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-cls 20825  df-nrm 21121

This theorem is referenced by:  isnrm3  21163