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Theorem clsconn 21233

Description: The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)

**Assertion**
Ref	Expression
clsconn	TopOn $/\$ $/\$ ↾_t Conn ↾_t Conn

Proof of Theorem clsconn Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll3 1102	. . . . 5 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ ↾_t Conn
2		simpll1 1100	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ TopOn
3		simpll2 1101	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
4		simplrl 800	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
5		simplrr 801	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
6		simprl1 1106	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
7		n0 3931	. . . . . . . . 9
8	6, 7	sylib 208	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
9	2	adantr 481	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ TopOn
10		topontop 20718	. . . . . . . . . 10 TopOn
11	9, 10	syl 17	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
12	3	adantr 481	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
13		toponuni 20719	. . . . . . . . . . 11 TopOn
14	9, 13	syl 17	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
15	12, 14	sseqtrd 3641	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
16		inss2 3834	. . . . . . . . . 10
17		simpr 477	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
18	16, 17	sseldi 3601	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
19	4	adantr 481	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
20		inss1 3833	. . . . . . . . . 10
21	20, 17	sseldi 3601	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
22		eqid 2622	. . . . . . . . . 10
23	22	clsndisj 20879	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
24	11, 15, 18, 19, 21, 23	syl32anc 1334	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
25	8, 24	exlimddv 1863	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
26		simprl2 1107	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
27		n0 3931	. . . . . . . . 9
28	26, 27	sylib 208	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
29	2	adantr 481	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ TopOn
30	29, 10	syl 17	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
31	3	adantr 481	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
32	29, 13	syl 17	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
33	31, 32	sseqtrd 3641	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
34		inss2 3834	. . . . . . . . . 10
35		simpr 477	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
36	34, 35	sseldi 3601	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
37	5	adantr 481	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
38		inss1 3833	. . . . . . . . . 10
39	38, 35	sseldi 3601	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
40	22	clsndisj 20879	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
41	30, 33, 36, 37, 39, 40	syl32anc 1334	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$
42	28, 41	exlimddv 1863	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
43		simprl3 1108	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$
44	2, 10	syl 17	. . . . . . . . . . . 12 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
45	2, 13	syl 17	. . . . . . . . . . . . 13 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
46	3, 45	sseqtrd 3641	. . . . . . . . . . . 12 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
47	22	sscls 20860	. . . . . . . . . . . 12 $/\$
48	44, 46, 47	syl2anc 693	. . . . . . . . . . 11 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
49	48	sscond 3747	. . . . . . . . . 10 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$ $\$
50	43, 49	sstrd 3613	. . . . . . . . 9 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$
51		ssv 3625	. . . . . . . . . 10
52		ssdif 3745	. . . . . . . . . 10 $\$ $\$
53	51, 52	ax-mp 5	. . . . . . . . 9 $\$ $\$
54	50, 53	syl6ss 3615	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $\$
55		disj2 4024	. . . . . . . 8 $\$
56	54, 55	sylibr 224	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
57		simprr 796	. . . . . . . 8 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
58	48, 57	sstrd 3613	. . . . . . 7 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$
59	2, 3, 4, 5, 25, 42, 56, 58	nconnsubb 21226	. . . . . 6 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ ↾_t Conn
60	59	expr 643	. . . . 5 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$ ↾_t Conn
61	1, 60	mt2d 131	. . . 4 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $/\$ $\$
62	61	ex 450	. . 3 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $/\$ $/\$ $\$
63	62	ralrimivva 2971	. 2 TopOn $/\$ $/\$ ↾_t Conn $/\$ $/\$ $\$
64		simp1 1061	. . 3 TopOn $/\$ $/\$ ↾_t Conn TopOn
65	13	sseq2d 3633	. . . . . . 7 TopOn
66	65	biimpa 501	. . . . . 6 TopOn $/\$
67	22	clsss3 20863	. . . . . . 7 $/\$
68	10, 67	sylan 488	. . . . . 6 TopOn $/\$
69	66, 68	syldan 487	. . . . 5 TopOn $/\$
70	13	adantr 481	. . . . 5 TopOn $/\$
71	69, 70	sseqtr4d 3642	. . . 4 TopOn $/\$
72	71	3adant3 1081	. . 3 TopOn $/\$ $/\$ ↾_t Conn
73		connsub 21224	. . 3 TopOn $/\$ ↾_t Conn $/\$ $/\$ $\$
74	64, 72, 73	syl2anc 693	. 2 TopOn $/\$ $/\$ ↾_t Conn ↾_t Conn $/\$ $/\$ $\$
75	63, 74	mpbird 247	1 TopOn $/\$ $/\$ ↾_t Conn ↾_t Conn

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wex 1704

wcel 1990

wne 2794

wral 2912

cvv 3200 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

cuni 4436

cfv 5888 (class class class)co 6650 ↾_t crest 16081

ctop 20698 TopOnctopon 20715

ccl 20822 Conncconn 21214

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-3or 1038 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 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-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-conn 21215

This theorem is referenced by: conncompcld 21237

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