Theorem discld 12142

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.

Step	Hyp	Ref	Expression
1		distop 12091	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		unipw 4097	. . . . . . 7 |- U. ~P A = A
3	2	eqcomi 2117	. . . . . 6 |- A = U. ~P A
4	3	iscld 12109	. . . . 5 |- ( ~P A e. Top -> ( x e. ( Clsd ` ~P A ) <-> ( x C_ A /\ ( A \ x ) e. ~P A ) ) )
5	1, 4	syl 14	. . . 4 |- ( A e. V -> ( x e. ( Clsd ` ~P A ) <-> ( x C_ A /\ ( A \ x ) e. ~P A ) ) )
6		difss 3166	. . . . . 6 |- ( A \ x ) C_ A
7		elpw2g 4039	. . . . . 6 |- ( A e. V -> ( ( A \ x ) e. ~P A <-> ( A \ x ) C_ A ) )
8	6, 7	mpbiri 167	. . . . 5 |- ( A e. V -> ( A \ x ) e. ~P A )
9	8	biantrud 300	. . . 4 |- ( A e. V -> ( x C_ A <-> ( x C_ A /\ ( A \ x ) e. ~P A ) ) )
10	5, 9	bitr4d 190	. . 3 |- ( A e. V -> ( x e. ( Clsd ` ~P A ) <-> x C_ A ) )
11		selpw 3481	. . 3 |- ( x e. ~P A <-> x C_ A )
12	10, 11	syl6bbr 197	. 2 |- ( A e. V -> ( x e. ( Clsd ` ~P A ) <-> x e. ~P A ) )
13	12	eqrdv 2111	1 |- ( A e. V -> ( Clsd ` ~P A ) = ~P A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   \ cdif 3032   C_ wss 3035   ~Pcpw 3474   U.cuni 3700   ` cfv 5079   Topctop 12001   Clsdccld 12098

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-top 12002  df-cld 12101

This theorem is referenced by:  sn0cld  12143