Theorem isnrm2 22417

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 22395	. . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2		nrmsep2 22415	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))
3	2	3exp2 1352	. . . . 5 ⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))))
4	3	impd 410	. . . 4 ⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
5	4	ralrimivv 3113	. . 3 ⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅)))
6	1, 5	jca 511	. 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑜 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅))))
7		simpl 482	. . 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 22092	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
10	9	adantr 480	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
11		ineq2 4137	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → (𝑐 ∩ 𝑑) = (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)))
12	11	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅))
13		ineq2 4137	. . . . . . . . . . . . . 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 628	. . . . . . . . . . . 12 ⊢ (𝑑 = (∪ 𝐽 ∖ 𝑥) → ((𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ 𝑑) = ∅) ↔ (𝑐 ⊆ 𝑜 ∧ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)))
16	15	rexbidv 3225	. . . . . . . . . . 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 344	. . . . . . . . . 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 3547	. . . . . . . . 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 4300	. . . . . . . . . 10 ⊢ ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
21	8	cldss 22088	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ ∪ 𝐽)
22	21	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝑐 ⊆ ∪ 𝐽)
23		df-ss 3900	. . . . . . . . . . . 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 3948	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ ∪ 𝐽) ⊆ 𝑥 ↔ 𝑐 ⊆ 𝑥))
26	20, 25	bitr3id 284	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ 𝑐 ⊆ 𝑥))
27		inssdif0 4300	. . . . . . . . . . . 12 ⊢ ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ (((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅)
28		simpll 763	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
29		elssuni 4868	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
30	8	clsss3 22118	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
31	28, 29, 30	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((cls‘𝐽)‘𝑜) ⊆ ∪ 𝐽)
32		df-ss 3900	. . . . . . . . . . . . . 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 3948	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ ∪ 𝐽) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
35	27, 34	bitr3id 284	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑐 ∈ (Clsd‘𝐽)) ∧ 𝑜 ∈ 𝐽) → ((((cls‘𝐽)‘𝑜) ∩ (∪ 𝐽 ∖ 𝑥)) = ∅ ↔ ((cls‘𝐽)‘𝑜) ⊆ 𝑥))
36	35	anbi2d 628	. . . . . . . . . 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 3224	. . . . . . . . 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 344	. . . . . . . 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 3102	. . . . . 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 3899	. . . . . . . . . 10 ⊢ (𝑐 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥) ↔ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑐 ∈ 𝒫 𝑥))
42		velpw 4535	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 𝑥 ↔ 𝑐 ⊆ 𝑥)
43	42	anbi2i 622	. . . . . . . . . 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 349	. . . . . . . 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 450	. . . . . . . 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 3088	. . . . . 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 3112	. . . 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 406	. . 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 22394	. . 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 582	. 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 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  Clsdccld 22075  clsccl 22077  Nrmcnrm 22369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-top 21951  df-cld 22078  df-cls 22080  df-nrm 22376

This theorem is referenced by:  isnrm3  22418