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Theorem iscld2 16871
Description: A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
iscld2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( X  \  S )  e.  J ) )

Proof of Theorem iscld2
StepHypRef Expression
1 iscld.1 . . 3  |-  X  = 
U. J
21iscld 16870 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
32baibd 875 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( X  \  S )  e.  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225    C_ wss 3228   U.cuni 3908   ` cfv 5337   Topctop 16737   Clsdccld 16859
This theorem is referenced by:  isopn2  16875  0cld  16881  uncld  16884  isclo  16930  cnclima  17103  ist1-2  17181  hausdiag  17445  qtopcld  17510  ufildr  17728  blcld  18153  icccld  18378  iocmnfcld  18380  zcld  18421  recld2  18422  kelac2  26486  stoweidlem50  27122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-top 16742  df-cld 16862
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