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Theorem discld 13529
Description: The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
discld  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )

Proof of Theorem discld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 distop 13478 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4217 . . . . . . 7  |-  U. ~P A  =  A
32eqcomi 2181 . . . . . 6  |-  A  = 
U. ~P A
43iscld 13496 . . . . 5  |-  ( ~P A  e.  Top  ->  ( x  e.  ( Clsd `  ~P A )  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
51, 4syl 14 . . . 4  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
6 difss 3261 . . . . . 6  |-  ( A 
\  x )  C_  A
7 elpw2g 4156 . . . . . 6  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
86, 7mpbiri 168 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
98biantrud 304 . . . 4  |-  ( A  e.  V  ->  (
x  C_  A  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
105, 9bitr4d 191 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  C_  A
) )
11 velpw 3582 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1210, 11bitr4di 198 . 2  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  e.  ~P A ) )
1312eqrdv 2175 1  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    \ cdif 3126    C_ wss 3129   ~Pcpw 3575   U.cuni 3809   ` cfv 5216   Topctop 13388   Clsdccld 13485
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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 13389  df-cld 13488
This theorem is referenced by:  sn0cld  13530
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