Theorem isnrm2 22509

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 22487	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep2 22507	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
3	2	3exp2 1353	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
4	3	impd 411	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
5	4	ralrimivv 3122	. . 3 ⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
6	1, 5	jca 512	. 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7		simpl 483	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))) → 𝐽 ∈ Top)
8		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
9	8	opncld 22184	. . . . . . . . . 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 4140	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (𝑐 ∩ 𝑑) = (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)))
12	11	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
13		ineq2 4140	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (((cls‘𝐽)‘𝑜) ∩ 𝑑) = (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)))
14	13	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅ ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
15	14	anbi2d 629	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)))
16	15	rexbidv 3226	. . . . . . . . . . 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 345	. . . . . . . . . 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 3557	. . . . . . . . 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 4303	. . . . . . . . . 10 ⊢ ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
21	8	cldss 22180	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ ∪ 𝐽)
22	21	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 ⊆ ∪ 𝐽)
23		df-ss 3904	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ∪ 𝐽 ↔ (𝑐 ∩ ∪ 𝐽) = 𝑐)
24	22, 23	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝑐 ∩ ∪ 𝐽) = 𝑐)
25	24	sseq1d 3952	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ 𝑐 ⊆ 𝑥))
26	20, 25	bitr3id 285	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ 𝑐 ⊆ 𝑥))
27		inssdif0 4303	. . . . . . . . . . . 12 ⊢ ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
28		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29		elssuni 4871	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
30	8	clsss3 22210	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
31	28, 29, 30	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
32		df-ss 3904	. . . . . . . . . . . . . 14 ⊢ (((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽 ↔ (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
33	31, 32	sylib 217	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → (((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) = ((cls‘𝐽)‘𝑜))
34	33	sseq1d 3952	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
35	27, 34	bitr3id 285	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
36	35	anbi2d 629	. . . . . . . . . 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 3225	. . . . . . . . 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 345	. . . . . . . 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 238	. . . . . . 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 3108	. . . . . 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 3903	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42		velpw 4538	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 𝑥 ↔ 𝑐 ⊆ 𝑥)
43	42	anbi2i 623	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
44	41, 43	bitri 274	. . . . . . . . 9 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ⊆ 𝑥))
45	44	imbi1i 350	. . . . . . . 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 451	. . . . . . . 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 274	. . . . . . 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 3090	. . . . . 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 251	. . . . 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 3106	. . . 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 407	. . 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 22486	. . 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 583	. 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 208	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  Clsdccld 22167  clsccl 22169  Nrmcnrm 22461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-cls 22172  df-nrm 22468

This theorem is referenced by:  isnrm3  22510