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Theorem discld 17158
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 17065 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4417 . . . . . . 7  |-  U. ~P A  =  A
32eqcomi 2442 . . . . . 6  |-  A  = 
U. ~P A
43iscld 17096 . . . . 5  |-  ( ~P A  e.  Top  ->  ( x  e.  ( Clsd `  ~P A )  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
51, 4syl 16 . . . 4  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
6 difss 3476 . . . . . 6  |-  ( A 
\  x )  C_  A
7 elpw2g 4366 . . . . . 6  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
86, 7mpbiri 226 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
98biantrud 495 . . . 4  |-  ( A  e.  V  ->  (
x  C_  A  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
105, 9bitr4d 249 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  C_  A
) )
11 vex 2961 . . . 4  |-  x  e. 
_V
1211elpw 3807 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1310, 12syl6bbr 256 . 2  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  e.  ~P A ) )
1413eqrdv 2436 1  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457   Topctop 16963   Clsdccld 17085
This theorem is referenced by:  sn0cld  17159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-top 16968  df-cld 17088
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