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Theorem opnbnd 30766
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 3873 . . . . 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 3663 . . . . 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 2431 . . . 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 223 . . 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 20003 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
87adantr 466 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  C_  A )
9 inssdif0 3868 . . . . . 6  |-  ( ( A  i^i  ( ( cls `  J ) `
 A ) ) 
C_  ( ( int `  J ) `  A
)  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) )
106sscls 20002 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( ( cls `  J ) `  A
) )
11 df-ss 3456 . . . . . . . . . 10  |-  ( A 
C_  ( ( cls `  J ) `  A
)  <->  ( A  i^i  ( ( cls `  J
) `  A )
)  =  A )
1210, 11sylib 199 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  i^i  (
( cls `  J
) `  A )
)  =  A )
1312eqcomd 2437 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  ( A  i^i  ( ( cls `  J
) `  A )
) )
14 eqimss 3522 . . . . . . . 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 3478 . . . . . . 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 473 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( cls `  J
) `  A )
)  C_  ( ( int `  J ) `  A ) )  ->  A  C_  ( ( int `  J ) `  A
) )
189, 17sylan2br 478 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  A  C_  ( ( int `  J
) `  A )
)
198, 18eqssd 3487 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( A  i^i  (
( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )  =  (/) )  ->  ( ( int `  J ) `
 A )  =  A )
2019ex 435 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/)  ->  ( ( int `  J ) `
 A )  =  A ) )
215, 20impbid 193 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  A
)  =  A  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) ) )
226isopn3 20013 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( ( int `  J
) `  A )  =  A ) )
236topbnd 30765 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
2423ineq2d 3670 . . 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 2431 . 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 288 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 187    /\ wa 370    = wceq 1437    e. wcel 1870    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767   U.cuni 4222   ` cfv 5601   Topctop 19848   intcnt 19963   clsccl 19964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-top 19852  df-cld 19965  df-ntr 19966  df-cls 19967
This theorem is referenced by:  cldbnd  30767
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