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Theorem discld 17116
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 17023 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4382 . . . . . . 7  |-  U. ~P A  =  A
32eqcomi 2416 . . . . . 6  |-  A  = 
U. ~P A
43iscld 17054 . . . . 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 3442 . . . . . 6  |-  ( A 
\  x )  C_  A
7 elpw2g 4331 . . . . . 6  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
86, 7mpbiri 225 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
98biantrud 494 . . . 4  |-  ( A  e.  V  ->  (
x  C_  A  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
105, 9bitr4d 248 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  C_  A
) )
11 vex 2927 . . . 4  |-  x  e. 
_V
1211elpw 3773 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1310, 12syl6bbr 255 . 2  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  e.  ~P A ) )
1413eqrdv 2410 1  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3285    C_ wss 3288   ~Pcpw 3767   U.cuni 3983   ` cfv 5421   Topctop 16921   Clsdccld 17043
This theorem is referenced by:  sn0cld  17117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-top 16926  df-cld 17046
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