isnrm3 - Metamath Proof Explorer

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Theorem isnrm3 21163

Description: A topological space is normal iff any two disjoint closed sets are separated by open sets. (Contributed by Mario Carneiro, 24-Aug-2015.)

**Assertion**
Ref	Expression
isnrm3	$/\$ $/\$ $/\$

Distinct variable groups:

**Proof of Theorem isnrm3**
Step	Hyp	Ref	Expression
1		nrmtop 21140	. . 3
2		nrmsep 21161	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	2	3exp2 1285	. . . . 5 $/\$ $/\$
4	3	impd 447	. . . 4 $/\$ $/\$ $/\$
5	4	ralrimivv 2970	. . 3 $/\$ $/\$
6	1, 5	jca 554	. 2 $/\$ $/\$ $/\$
7		simpl 473	. . 3 $/\$ $/\$ $/\$
8		simpr1 1067	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr2 1068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
10		sslin 3839	. . . . . . . . . . . . 13
11	9, 10	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12		simplll 798	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
13		simplr 792	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
14		eqid 2622	. . . . . . . . . . . . . . 15
15	14	opncld 20837	. . . . . . . . . . . . . 14 $/\$ $\$
16	12, 13, 15	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
17		simpr3 1069	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
18		simpllr 799	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
19		elssuni 4467	. . . . . . . . . . . . . . 15
20		reldisj 4020	. . . . . . . . . . . . . . 15 $\$
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
22	17, 21	mpbid 222	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
23	14	clsss2 20876	. . . . . . . . . . . . . 14 $\$ $/\$ $\$ $\$
24		ssdifin0 4050	. . . . . . . . . . . . . 14 $\$
25	23, 24	syl 17	. . . . . . . . . . . . 13 $\$ $/\$ $\$
26	16, 22, 25	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
27		sseq0 3975	. . . . . . . . . . . 12 $/\$
28	11, 26, 27	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
29	8, 28	jca 554	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	ex 450	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
31	30	rexlimdva 3031	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	31	reximdva 3017	. . . . . . 7 $/\$ $/\$ $/\$
33	32	imim2d 57	. . . . . 6 $/\$ $/\$ $/\$
34	33	ralimdv 2963	. . . . 5 $/\$ $/\$ $/\$
35	34	ralimdv 2963	. . . 4 $/\$ $/\$ $/\$
36	35	imp 445	. . 3 $/\$ $/\$ $/\$ $/\$
37		isnrm2 21162	. . 3 $/\$ $/\$
38	7, 36, 37	sylanbrc 698	. 2 $/\$ $/\$ $/\$
39	6, 38	impbii 199	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913 $\$ cdif 3571

cin 3573

wss 3574

c0 3915

cuni 4436

cfv 5888

ctop 20698

ccld 20820

ccl 20822

cnrm 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: metnrm 22665

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