MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difopn Unicode version

Theorem difopn 16988
Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
difopn  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )

Proof of Theorem difopn
StepHypRef Expression
1 elssuni 3957 . . . . . 6  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 iscld.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6sseqr 3311 . . . . 5  |-  ( A  e.  J  ->  A  C_  X )
43adantr 451 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  C_  X )
5 df-ss 3252 . . . 4  |-  ( A 
C_  X  <->  ( A  i^i  X )  =  A )
64, 5sylib 188 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  X
)  =  A )
76difeq1d 3380 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  =  ( A  \  B ) )
8 indif2 3500 . . 3  |-  ( A  i^i  ( X  \  B ) )  =  ( ( A  i^i  X )  \  B )
9 cldrcl 16980 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
109adantl 452 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  J  e.  Top )
11 simpl 443 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  e.  J )
122cldopn 16985 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1312adantl 452 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( X  \  B
)  e.  J )
14 inopn 16862 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  J  /\  ( X  \  B )  e.  J )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
1510, 11, 13, 14syl3anc 1183 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
168, 15syl5eqelr 2451 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  e.  J )
177, 16eqeltrrd 2441 1  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    \ cdif 3235    i^i cin 3237    C_ wss 3238   U.cuni 3929   ` cfv 5358   Topctop 16848   Clsdccld 16970
This theorem is referenced by:  bcthlem5  18965  cldssbrsiga  24006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fn 5361  df-fv 5366  df-top 16853  df-cld 16973
  Copyright terms: Public domain W3C validator