Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnbnd Unicode version

Theorem opnbnd 26243
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 3526 . . . . 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 3363 . . . . 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 2291 . . . 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 16794 . . . . . 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 3521 . . . . . 6  |-  ( ( A  i^i  ( ( cls `  J ) `
 A ) ) 
C_  ( ( int `  J ) `  A
)  <->  ( A  i^i  ( ( ( cls `  J ) `  A
)  \  ( ( int `  J ) `  A ) ) )  =  (/) )
106sscls 16793 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  ( ( cls `  J ) `  A
) )
11 df-ss 3166 . . . . . . . . . 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 2288 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  =  ( A  i^i  ( ( cls `  J
) `  A )
) )
14 eqimss 3230 . . . . . . . 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 3187 . . . . . . 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 3196 . . . 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 16803 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  J  <->  ( ( int `  J
) `  A )  =  A ) )
236topbnd 26242 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
2423ineq2d 3370 . . 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 2291 . 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 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754   clsccl 16755
This theorem is referenced by:  cldbnd  26244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758
  Copyright terms: Public domain W3C validator