elcls3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elcls3		Structured version Visualization version Unicode version

Theorem elcls3 20887

Description: Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.)

**Hypotheses**
Ref	Expression
elcls3.1
elcls3.2
elcls3.3
elcls3.4
elcls3.5

**Assertion**
Ref	Expression
elcls3

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

)

Proof of Theorem elcls3 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcls3.1	. . . 4
2		elcls3.3	. . . . 5
3		tgcl 20773	. . . . 5
4	2, 3	syl 17	. . . 4
5	1, 4	eqeltrd 2701	. . 3
6		elcls3.4	. . . 4
7		elcls3.2	. . . 4
8	6, 7	sseqtrd 3641	. . 3
9		elcls3.5	. . . 4
10	9, 7	eleqtrd 2703	. . 3
11		eqid 2622	. . . 4
12	11	elcls 20877	. . 3 $/\$ $/\$
13	5, 8, 10, 12	syl3anc 1326	. 2
14		bastg 20770	. . . . . . . . 9
15	2, 14	syl 17	. . . . . . . 8
16	15, 1	sseqtr4d 3642	. . . . . . 7
17	16	sseld 3602	. . . . . 6
18	17	imim1d 82	. . . . 5
19	18	ralimdv2 2961	. . . 4
20		eleq2 2690	. . . . . 6
21		ineq1 3807	. . . . . . 7
22	21	neeq1d 2853	. . . . . 6
23	20, 22	imbi12d 334	. . . . 5
24	23	cbvralv 3171	. . . 4
25	19, 24	syl6ib 241	. . 3
26		simprl 794	. . . . . . . 8 $/\$ $/\$ $/\$
27	1	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$
28	26, 27	eleqtrd 2703	. . . . . . 7 $/\$ $/\$ $/\$
29		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$
30		tg2 20769	. . . . . . 7 $/\$ $/\$
31	28, 29, 30	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$
32		eleq2 2690	. . . . . . . . . . . . . 14
33		ineq1 3807	. . . . . . . . . . . . . . 15
34	33	neeq1d 2853	. . . . . . . . . . . . . 14
35	32, 34	imbi12d 334	. . . . . . . . . . . . 13
36	35	rspccva 3308	. . . . . . . . . . . 12 $/\$
37	36	imp 445	. . . . . . . . . . 11 $/\$ $/\$
38		ssdisj 4026	. . . . . . . . . . . . 13 $/\$
39	38	ex 450	. . . . . . . . . . . 12
40	39	necon3d 2815	. . . . . . . . . . 11
41	37, 40	syl5com 31	. . . . . . . . . 10 $/\$ $/\$
42	41	exp31 630	. . . . . . . . 9
43	42	imp4a 614	. . . . . . . 8 $/\$
44	43	rexlimdv 3030	. . . . . . 7 $/\$
45	44	ad2antlr 763	. . . . . 6 $/\$ $/\$ $/\$ $/\$
46	31, 45	mpd 15	. . . . 5 $/\$ $/\$ $/\$
47	46	exp43 640	. . . 4
48	47	ralrimdv 2968	. . 3
49	25, 48	impbid 202	. 2
50	13, 49	bitrd 268	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

cin 3573

wss 3574

c0 3915

cuni 4436

cfv 5888

ctg 16098

ctop 20698

ctb 20749

ccl 20822

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-topgen 16104 df-top 20699 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825

This theorem is referenced by: 2ndcsep 21262 ptclsg 21418 qdensere 22573

W3C validator