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Theorem difopn 17061
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 4011 . . . . . 6  |-  ( A  e.  J  ->  A  C_ 
U. J )
2 iscld.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6sseqr 3363 . . . . 5  |-  ( A  e.  J  ->  A  C_  X )
43adantr 452 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  C_  X )
5 df-ss 3302 . . . 4  |-  ( A 
C_  X  <->  ( A  i^i  X )  =  A )
64, 5sylib 189 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  X
)  =  A )
76difeq1d 3432 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  =  ( A  \  B ) )
8 indif2 3552 . . 3  |-  ( A  i^i  ( X  \  B ) )  =  ( ( A  i^i  X )  \  B )
9 cldrcl 17053 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  J  e.  Top )
109adantl 453 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  J  e.  Top )
11 simpl 444 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  ->  A  e.  J )
122cldopn 17058 . . . . 5  |-  ( B  e.  ( Clsd `  J
)  ->  ( X  \  B )  e.  J
)
1312adantl 453 . . . 4  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( X  \  B
)  e.  J )
14 inopn 16935 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  J  /\  ( X  \  B )  e.  J )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
1510, 11, 13, 14syl3anc 1184 . . 3  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  i^i  ( X  \  B ) )  e.  J )
168, 15syl5eqelr 2497 . 2  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( ( A  i^i  X )  \  B )  e.  J )
177, 16eqeltrrd 2487 1  |-  ( ( A  e.  J  /\  B  e.  ( Clsd `  J ) )  -> 
( A  \  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3285    i^i cin 3287    C_ wss 3288   U.cuni 3983   ` cfv 5421   Topctop 16921   Clsdccld 17043
This theorem is referenced by:  bcthlem5  19242  cldssbrsiga  24502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-top 16926  df-cld 17046
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