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Theorem iscld 13534
Description: The predicate "the class  S is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
iscld  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )

Proof of Theorem iscld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21cldval 13530 . . . 4  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
32eleq2d 2247 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  { x  e.  ~P X  |  ( X  \  x )  e.  J } ) )
4 difeq2 3247 . . . . 5  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
54eleq1d 2246 . . . 4  |-  ( x  =  S  ->  (
( X  \  x
)  e.  J  <->  ( X  \  S )  e.  J
) )
65elrab 2893 . . 3  |-  ( S  e.  { x  e. 
~P X  |  ( X  \  x )  e.  J }  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) )
73, 6bitrdi 196 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) ) )
81topopn 13439 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4156 . . . 4  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
108, 9syl 14 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1110anbi1d 465 . 2  |-  ( J  e.  Top  ->  (
( S  e.  ~P X  /\  ( X  \  S )  e.  J
)  <->  ( S  C_  X  /\  ( X  \  S )  e.  J
) ) )
127, 11bitrd 188 1  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {crab 2459    \ cdif 3126    C_ wss 3129   ~Pcpw 3575   U.cuni 3809   ` cfv 5216   Topctop 13428   Clsdccld 13523
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-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-top 13429  df-cld 13526
This theorem is referenced by:  iscld2  13535  cldss  13536  cldopn  13538  topcld  13540  discld  13567
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