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Theorem discld 18692
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 18599 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4541 . . . . . . 7  |-  U. ~P A  =  A
32eqcomi 2446 . . . . . 6  |-  A  = 
U. ~P A
43iscld 18630 . . . . 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 3482 . . . . . 6  |-  ( A 
\  x )  C_  A
7 elpw2g 4454 . . . . . 6  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
86, 7mpbiri 233 . . . . 5  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
98biantrud 507 . . . 4  |-  ( A  e.  V  ->  (
x  C_  A  <->  ( x  C_  A  /\  ( A 
\  x )  e. 
~P A ) ) )
105, 9bitr4d 256 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  C_  A
) )
11 selpw 3866 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1210, 11syl6bbr 263 . 2  |-  ( A  e.  V  ->  (
x  e.  ( Clsd `  ~P A )  <->  x  e.  ~P A ) )
1312eqrdv 2440 1  |-  ( A  e.  V  ->  ( Clsd `  ~P A )  =  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3324    C_ wss 3327   ~Pcpw 3859   U.cuni 4090   ` cfv 5417   Topctop 18497   Clsdccld 18619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-top 18502  df-cld 18622
This theorem is referenced by:  sn0cld  18693
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