Theorem opnbnd 15409

Description: A set is open iff it is disjoint from its boundary.

Hypothesis

Ref	Expression
opnbnd.1	|- X = U.J

Assertion

Ref	Expression
opnbnd	|- ((J e. Top /\ A C_ X) -> (A e. J <-> (A i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/)))

Proof of Theorem opnbnd

Step	Hyp	Ref	Expression
1		ineq1 2789	. . . . 5 |- (((int` J)` A) = A -> (((int` J)` A) i^i (((cls` J)` A) \ ((int` J)` A))) = (A i^i (((cls` J)` A) \ ((int` J)` A))))
2	1	eqeq1d 1892	. . . 4 |- (((int` J)` A) = A -> ((((int` J)` A) i^i (((cls` J)` A) \ ((int` J)` A))) = (/) <-> (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)))
3		difdisj 2945	. . . . 5 |- (((int` J)` A) i^i (((cls` J)` A) \ ((int` J)` A))) = (/)
4	3	a1i 8	. . . 4 |- ((J e. Top /\ A C_ X) -> (((int` J)` A) i^i (((cls` J)` A) \ ((int` J)` A))) = (/))
5	2, 4	syl5cbi 226	. . 3 |- ((J e. Top /\ A C_ X) -> (((int` J)` A) = A -> (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)))
6		opnbnd.1	. . . . . . 7 |- X = U.J
7	6	ntrss2 8966	. . . . . 6 |- ((J e. Top /\ A C_ X) -> ((int` J)` A) C_ A)
8	7	adantr 425	. . . . 5 |- (((J e. Top /\ A C_ X) /\ (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)) -> ((int` J)` A) C_ A)
9		sstr 2625	. . . . . . 7 |- ((A C_ (A i^i ((cls` J)` A)) /\ (A i^i ((cls` J)` A)) C_ ((int` J)` A)) -> A C_ ((int` J)` A))
10	6	sscls 8965	. . . . . . . . . 10 |- ((J e. Top /\ A C_ X) -> A C_ ((cls` J)` A))
11		df-ss 2605	. . . . . . . . . 10 |- (A C_ ((cls` J)` A) <-> (A i^i ((cls` J)` A)) = A)
12	10, 11	sylib 215	. . . . . . . . 9 |- ((J e. Top /\ A C_ X) -> (A i^i ((cls` J)` A)) = A)
13	12	eqcomd 1889	. . . . . . . 8 |- ((J e. Top /\ A C_ X) -> A = (A i^i ((cls` J)` A)))
14		eqimss 2665	. . . . . . . 8 |- (A = (A i^i ((cls` J)` A)) -> A C_ (A i^i ((cls` J)` A)))
15	13, 14	syl 12	. . . . . . 7 |- ((J e. Top /\ A C_ X) -> A C_ (A i^i ((cls` J)` A)))
16	9, 15	sylan 497	. . . . . 6 |- (((J e. Top /\ A C_ X) /\ (A i^i ((cls` J)` A)) C_ ((int` J)` A)) -> A C_ ((int` J)` A))
17		inssdif0 2940	. . . . . 6 |- ((A i^i ((cls` J)` A)) C_ ((int` J)` A) <-> (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/))
18	16, 17	sylan2br 502	. . . . 5 |- (((J e. Top /\ A C_ X) /\ (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)) -> A C_ ((int` J)` A))
19	8, 18	eqssd 2633	. . . 4 |- (((J e. Top /\ A C_ X) /\ (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)) -> ((int` J)` A) = A)
20	19	ex 402	. . 3 |- ((J e. Top /\ A C_ X) -> ((A i^i (((cls` J)` A) \ ((int` J)` A))) = (/) -> ((int` J)` A) = A))
21	5, 20	impbid 574	. 2 |- ((J e. Top /\ A C_ X) -> (((int` J)` A) = A <-> (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)))
22	6	isopn3 8973	. 2 |- ((J e. Top /\ A C_ X) -> (A e. J <-> ((int` J)` A) = A))
23	6	topbnd 15408	. . . 4 |- ((J e. Top /\ A C_ X) -> (((cls` J)` A) i^i ((cls` J)` (X \ A))) = (((cls` J)` A) \ ((int` J)` A)))
24	23	ineq2d 2796	. . 3 |- ((J e. Top /\ A C_ X) -> (A i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (A i^i (((cls` J)` A) \ ((int` J)` A))))
25	24	eqeq1d 1892	. 2 |- ((J e. Top /\ A C_ X) -> ((A i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/) <-> (A i^i (((cls` J)` A) \ ((int` J)` A))) = (/)))
26	21, 22, 25	3bitr4d 609	1 |- ((J e. Top /\ A C_ X) -> (A e. J <-> (A i^i (((cls` J)` A) i^i ((cls` J)` (X \ A)))) = (/)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  ` cfv 3998  Topctop 8857  intcnt 8937  clsccl 8938

This theorem is referenced by:  cldbnd 15410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941

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