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Theorem lpcls 21168

Description: The limit points of the closure of a subset are the same as the limit points of the set in a T₁ space. (Contributed by Mario Carneiro, 26-Dec-2016.)

**Hypothesis**
Ref	Expression
lpcls.1

**Assertion**
Ref	Expression
lpcls	$/\$

Proof of Theorem lpcls Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		t1top 21134	. . . . . . 7
2		lpcls.1	. . . . . . . . . 10
3	2	clsss3 20863	. . . . . . . . 9 $/\$
4	3	ssdifssd 3748	. . . . . . . 8 $/\$ $\$
5	2	clsss3 20863	. . . . . . . 8 $/\$ $\$ $\$
6	4, 5	syldan 487	. . . . . . 7 $/\$ $\$
7	1, 6	sylan 488	. . . . . 6 $/\$ $\$
8	7	sseld 3602	. . . . 5 $/\$ $\$
9		ssdifss 3741	. . . . . . . . . . 11 $\$
10	2	clscld 20851	. . . . . . . . . . 11 $/\$ $\$ $\$
11	1, 9, 10	syl2an 494	. . . . . . . . . 10 $/\$ $\$
12	11	adantr 481	. . . . . . . . 9 $/\$ $/\$ $\$
13	2	t1sncld 21130	. . . . . . . . . . . . 13 $/\$
14	13	adantlr 751	. . . . . . . . . . . 12 $/\$ $/\$
15		uncld 20845	. . . . . . . . . . . 12 $/\$ $\$ $\$
16	14, 12, 15	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $\$
17	2	sscls 20860	. . . . . . . . . . . . . 14 $/\$ $\$ $\$ $\$
18	1, 9, 17	syl2an 494	. . . . . . . . . . . . 13 $/\$ $\$ $\$
19		ssundif 4052	. . . . . . . . . . . . 13 $\$ $\$ $\$
20	18, 19	sylibr 224	. . . . . . . . . . . 12 $/\$ $\$
21	20	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $\$
22	2	clsss2 20876	. . . . . . . . . . 11 $\$ $/\$ $\$ $\$
23	16, 21, 22	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $\$
24		ssundif 4052	. . . . . . . . . 10 $\$ $\$ $\$
25	23, 24	sylib 208	. . . . . . . . 9 $/\$ $/\$ $\$ $\$
26	2	clsss2 20876	. . . . . . . . 9 $\$ $/\$ $\$ $\$ $\$ $\$
27	12, 25, 26	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $\$ $\$
28	27	sseld 3602	. . . . . . 7 $/\$ $/\$ $\$ $\$
29	28	ex 450	. . . . . 6 $/\$ $\$ $\$
30	29	com23 86	. . . . 5 $/\$ $\$ $\$
31	8, 30	mpdd 43	. . . 4 $/\$ $\$ $\$
32	1	adantr 481	. . . . . 6 $/\$
33	1, 3	sylan 488	. . . . . . 7 $/\$
34	33	ssdifssd 3748	. . . . . 6 $/\$ $\$
35	2	sscls 20860	. . . . . . . 8 $/\$
36	1, 35	sylan 488	. . . . . . 7 $/\$
37	36	ssdifd 3746	. . . . . 6 $/\$ $\$ $\$
38	2	clsss 20858	. . . . . 6 $/\$ $\$ $/\$ $\$ $\$ $\$ $\$
39	32, 34, 37, 38	syl3anc 1326	. . . . 5 $/\$ $\$ $\$
40	39	sseld 3602	. . . 4 $/\$ $\$ $\$
41	31, 40	impbid 202	. . 3 $/\$ $\$ $\$
42	2	islp 20944	. . . . 5 $/\$ $\$
43	3, 42	syldan 487	. . . 4 $/\$ $\$
44	1, 43	sylan 488	. . 3 $/\$ $\$
45	2	islp 20944	. . . 4 $/\$ $\$
46	1, 45	sylan 488	. . 3 $/\$ $\$
47	41, 44, 46	3bitr4d 300	. 2 $/\$
48	47	eqrdv 2620	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990 $\$ cdif 3571

cun 3572

wss 3574

csn 4177

cuni 4436

cfv 5888

ctop 20698

ccld 20820

ccl 20822

clp 20938

ct1 21111

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-lp 20940 df-t1 21118

This theorem is referenced by: perfcls 21169

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