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Theorem opnbnd 30070
Description: A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
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
StepHypRef Expression
1 disjdif 3905 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)
21a1i 11 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  i^i  ( (
( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )
3 ineq1 3698 . . . . 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 ) ) ) )
43eqeq1d 2469 . . . 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 ) ) )  =  (/) ) )
52, 4syl5ibcom 220 . . 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
76ntrss2 19426 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
87adantr 465 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  C_  A )
9 inssdif0 3900 . . . . . 6  |-  ( ( A  i^i  ( ( cls `  J ) `
 A ) ) 
C_  ( ( int `  J ) `  A
)  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) )
106sscls 19425 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( ( cls `  J ) `  A
) )
11 df-ss 3495 . . . . . . . . . 10  |-  ( A 
C_  ( ( cls `  J ) `  A
)  <->  ( A  i^i  ( ( cls `  J
) `  A )
)  =  A )
1210, 11sylib 196 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  i^i  (
( cls `  J
) `  A )
)  =  A )
1312eqcomd 2475 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  ( A  i^i  ( ( cls `  J
) `  A )
) )
14 eqimss 3561 . . . . . . . 8  |-  ( A  =  ( A  i^i  ( ( cls `  J
) `  A )
)  ->  A  C_  ( A  i^i  ( ( cls `  J ) `  A
) ) )
1513, 14syl 16 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( A  i^i  ( ( cls `  J
) `  A )
) )
16 sstr 3517 . . . . . . 7  |-  ( ( A  C_  ( A  i^i  ( ( cls `  J
) `  A )
)  /\  ( A  i^i  ( ( cls `  J
) `  A )
)  C_  ( ( int `  J ) `  A ) )  ->  A  C_  ( ( int `  J ) `  A
) )
1715, 16sylan 471 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( cls `  J
) `  A )
)  C_  ( ( int `  J ) `  A ) )  ->  A  C_  ( ( int `  J ) `  A
) )
189, 17sylan2br 476 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  A  C_  ( ( int `  J
) `  A )
)
198, 18eqssd 3526 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  =  A )
2019ex 434 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)  ->  ( ( int `  J ) `
 A )  =  A ) )
215, 20impbid 191 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  =  A  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) ) )
226isopn3 19435 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( ( int `  J
) `  A )  =  A ) )
236topbnd 30069 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
2423ineq2d 3705 . . 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 )
) ) )
2524eqeq1d 2469 . 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
) ) )  =  (/) ) )
2621, 22, 253bitr4d 285 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( A  i^i  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   U.cuni 4251   ` cfv 5594   Topctop 19263   intcnt 19386   clsccl 19387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390
This theorem is referenced by:  cldbnd  30071
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