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Theorem opnbnd 26346
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 3539 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)
21a1i 10 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  i^i  ( (
( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )
3 ineq1 3376 . . . . 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 2304 . . . 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 211 . . 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 16810 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
87adantr 451 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  C_  A )
9 inssdif0 3534 . . . . . 6  |-  ( ( A  i^i  ( ( cls `  J ) `
 A ) ) 
C_  ( ( int `  J ) `  A
)  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) )
106sscls 16809 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( ( cls `  J ) `  A
) )
11 df-ss 3179 . . . . . . . . . 10  |-  ( A 
C_  ( ( cls `  J ) `  A
)  <->  ( A  i^i  ( ( cls `  J
) `  A )
)  =  A )
1210, 11sylib 188 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  i^i  (
( cls `  J
) `  A )
)  =  A )
1312eqcomd 2301 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  ( A  i^i  ( ( cls `  J
) `  A )
) )
14 eqimss 3243 . . . . . . . 8  |-  ( A  =  ( A  i^i  ( ( cls `  J
) `  A )
)  ->  A  C_  ( A  i^i  ( ( cls `  J ) `  A
) ) )
1513, 14syl 15 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( A  i^i  ( ( cls `  J
) `  A )
) )
16 sstr 3200 . . . . . . 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 457 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( cls `  J
) `  A )
)  C_  ( ( int `  J ) `  A ) )  ->  A  C_  ( ( int `  J ) `  A
) )
189, 17sylan2br 462 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  A  C_  ( ( int `  J
) `  A )
)
198, 18eqssd 3209 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  =  A )
2019ex 423 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)  ->  ( ( int `  J ) `
 A )  =  A ) )
215, 20impbid 183 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  =  A  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) ) )
226isopn3 16819 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( ( int `  J
) `  A )  =  A ) )
236topbnd 26345 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
2423ineq2d 3383 . . 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 2304 . 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 276 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770   clsccl 16771
This theorem is referenced by:  cldbnd  26347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774
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