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Theorem iscld2 18607
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 18606 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
32baibd 900 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( X  \  S )  e.  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3320    C_ wss 3323   U.cuni 4086   ` cfv 5413   Topctop 18473   Clsdccld 18595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-top 18478  df-cld 18598
This theorem is referenced by:  isopn2  18611  0cld  18617  uncld  18620  isclo  18666  cnclima  18847  ist1-2  18926  hausdiag  19193  qtopcld  19261  ufildr  19479  blcld  20055  icccld  20321  iocmnfcld  20323  zcld  20365  recld2  20366  dvtanlem  28394  kelac2  29371  stoweidlem50  29798
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