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Theorem discld 12232
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 12181 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4109 . . . . . . 7  |-  U. ~P A  =  A
32eqcomi 2121 . . . . . 6  |-  A  = 
U. ~P A
43iscld 12199 . . . . 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 3172 . . . . . 6  |-  ( A 
\  x )  C_  A
7 elpw2g 4051 . . . . . 6  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
86, 7mpbiri 167 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
98biantrud 302 . . . 4  |-  ( A  e.  V  ->  (
x  C_  A  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
105, 9bitr4d 190 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  C_  A
) )
11 velpw 3487 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1210, 11syl6bbr 197 . 2  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  e.  ~P A ) )
1312eqrdv 2115 1  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465    \ cdif 3038    C_ wss 3041   ~Pcpw 3480   U.cuni 3706   ` cfv 5093   Topctop 12091   Clsdccld 12188
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-top 12092  df-cld 12191
This theorem is referenced by:  sn0cld  12233
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