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Theorem difopn 17103
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 4045 . . . . . 6  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 iscld.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6sseqr 3397 . . . . 5  |-  ( A  e.  J  ->  A  C_  X )
43adantr 453 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  C_  X )
5 df-ss 3336 . . . 4  |-  ( A 
C_  X  <->  ( A  i^i  X )  =  A )
64, 5sylib 190 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  X
)  =  A )
76difeq1d 3466 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  =  ( A  \  B ) )
8 indif2 3586 . . 3  |-  ( A  i^i  ( X  \  B ) )  =  ( ( A  i^i  X )  \  B )
9 cldrcl 17095 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
109adantl 454 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  J  e.  Top )
11 simpl 445 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  e.  J )
122cldopn 17100 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1312adantl 454 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( X  \  B
)  e.  J )
14 inopn 16977 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  J  /\  ( X  \  B )  e.  J )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
1510, 11, 13, 14syl3anc 1185 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
168, 15syl5eqelr 2523 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  e.  J )
177, 16eqeltrrd 2513 1  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322   U.cuni 4017   ` cfv 5457   Topctop 16963   Clsdccld 17085
This theorem is referenced by:  bcthlem5  19286  cldssbrsiga  24546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-top 16968  df-cld 17088
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