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Theorem opnbnd 30993
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 3841 . . . . 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 3629 . . . . 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 2455 . . . 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 224 . . 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 20084 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
87adantr 467 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  C_  A )
9 inssdif0 3836 . . . . . 6  |-  ( ( A  i^i  ( ( cls `  J ) `
 A ) ) 
C_  ( ( int `  J ) `  A
)  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) )
106sscls 20083 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( ( cls `  J ) `  A
) )
11 df-ss 3420 . . . . . . . . . 10  |-  ( A 
C_  ( ( cls `  J ) `  A
)  <->  ( A  i^i  ( ( cls `  J
) `  A )
)  =  A )
1210, 11sylib 200 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  i^i  (
( cls `  J
) `  A )
)  =  A )
1312eqcomd 2459 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  ( A  i^i  ( ( cls `  J
) `  A )
) )
14 eqimss 3486 . . . . . . . 8  |-  ( A  =  ( A  i^i  ( ( cls `  J
) `  A )
)  ->  A  C_  ( A  i^i  ( ( cls `  J ) `  A
) ) )
1513, 14syl 17 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( A  i^i  ( ( cls `  J
) `  A )
) )
16 sstr 3442 . . . . . . 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 474 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( cls `  J
) `  A )
)  C_  ( ( int `  J ) `  A ) )  ->  A  C_  ( ( int `  J ) `  A
) )
189, 17sylan2br 479 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  A  C_  ( ( int `  J
) `  A )
)
198, 18eqssd 3451 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  =  A )
2019ex 436 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)  ->  ( ( int `  J ) `
 A )  =  A ) )
215, 20impbid 194 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  =  A  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) ) )
226isopn3 20094 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( ( int `  J
) `  A )  =  A ) )
236topbnd 30992 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
2423ineq2d 3636 . . 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 2455 . 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 289 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 188    /\ wa 371    = wceq 1446    e. wcel 1889    \ cdif 3403    i^i cin 3405    C_ wss 3406   (/)c0 3733   U.cuni 4201   ` cfv 5585   Topctop 19929   intcnt 20044   clsccl 20045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-top 19933  df-cld 20046  df-ntr 20047  df-cls 20048
This theorem is referenced by:  cldbnd  30994
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