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Theorem difopn 13241
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 3835 . . . . . 6  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 iscld.1 . . . . . 6  |-  X  = 
U. J
31, 2sseqtrrdi 3204 . . . . 5  |-  ( A  e.  J  ->  A  C_  X )
43adantr 276 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  C_  X )
5 df-ss 3142 . . . 4  |-  ( A 
C_  X  <->  ( A  i^i  X )  =  A )
64, 5sylib 122 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  X
)  =  A )
76difeq1d 3252 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  =  ( A  \  B ) )
8 indif2 3379 . . 3  |-  ( A  i^i  ( X  \  B ) )  =  ( ( A  i^i  X )  \  B )
9 cldrcl 13235 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
109adantl 277 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  J  e.  Top )
11 simpl 109 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  e.  J )
122cldopn 13240 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1312adantl 277 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( X  \  B
)  e.  J )
14 inopn 13134 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  J  /\  ( X  \  B )  e.  J )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
1510, 11, 13, 14syl3anc 1238 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
168, 15eqeltrrid 2265 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  e.  J )
177, 16eqeltrrd 2255 1  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    \ cdif 3126    i^i cin 3128    C_ wss 3129   U.cuni 3807   ` cfv 5211   Topctop 13128   Clsdccld 13225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-top 13129  df-cld 13228
This theorem is referenced by: (None)
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