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Theorem difopn 12314
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 3771 . . . . . 6  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 iscld.1 . . . . . 6  |-  X  = 
U. J
31, 2sseqtrrdi 3150 . . . . 5  |-  ( A  e.  J  ->  A  C_  X )
43adantr 274 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  C_  X )
5 df-ss 3088 . . . 4  |-  ( A 
C_  X  <->  ( A  i^i  X )  =  A )
64, 5sylib 121 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  X
)  =  A )
76difeq1d 3197 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  =  ( A  \  B ) )
8 indif2 3324 . . 3  |-  ( A  i^i  ( X  \  B ) )  =  ( ( A  i^i  X )  \  B )
9 cldrcl 12308 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
109adantl 275 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  J  e.  Top )
11 simpl 108 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  e.  J )
122cldopn 12313 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1312adantl 275 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( X  \  B
)  e.  J )
14 inopn 12207 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  J  /\  ( X  \  B )  e.  J )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
1510, 11, 13, 14syl3anc 1217 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
168, 15eqeltrrid 2228 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  e.  J )
177, 16eqeltrrd 2218 1  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481    \ cdif 3072    i^i cin 3074    C_ wss 3075   U.cuni 3743   ` cfv 5130   Topctop 12201   Clsdccld 12298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-top 12202  df-cld 12301
This theorem is referenced by: (None)
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