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Theorem iscld 12897
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 12893 . . . 4  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
32eleq2d 2240 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  { x  e.  ~P X  |  ( X  \  x )  e.  J } ) )
4 difeq2 3239 . . . . 5  |-  ( x  =  S  ->  ( X  \  x )  =  ( X  \  S
) )
54eleq1d 2239 . . . 4  |-  ( x  =  S  ->  (
( X  \  x
)  e.  J  <->  ( X  \  S )  e.  J
) )
65elrab 2886 . . 3  |-  ( S  e.  { x  e. 
~P X  |  ( X  \  x )  e.  J }  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) )
73, 6bitrdi 195 . 2  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  e.  ~P X  /\  ( X  \  S )  e.  J ) ) )
81topopn 12800 . . . 4  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4142 . . . 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 462 . 2  |-  ( J  e.  Top  ->  (
( S  e.  ~P X  /\  ( X  \  S )  e.  J
)  <->  ( S  C_  X  /\  ( X  \  S )  e.  J
) ) )
127, 11bitrd 187 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 103    <-> wb 104    = wceq 1348    e. wcel 2141   {crab 2452    \ cdif 3118    C_ wss 3121   ~Pcpw 3566   U.cuni 3796   ` cfv 5198   Topctop 12789   Clsdccld 12886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-top 12790  df-cld 12889
This theorem is referenced by:  iscld2  12898  cldss  12899  cldopn  12901  topcld  12903  discld  12930
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