Theorem discld 15127

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 15076	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		unipw 4338	. . . . . . 7 |- U. ~P A = A
3	2	eqcomi 2238	. . . . . 6 |- A = U. ~P A
4	3	iscld 15094	. . . . 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 3349	. . . . . 6 |- ( A \ x ) C_ A
7		elpw2g 4273	. . . . . 6 |- ( A e. V -> ( ( A \ x ) e. ~P A <-> ( A \ x ) C_ A ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A e. V -> ( A \ x ) e. ~P A )
9	8	biantrud 304	. . . 4 |- ( A e. V -> ( x C_ A <-> ( x C_ A /\ ( A \ x ) e. ~P A ) ) )
10	5, 9	bitr4d 191	. . 3 |- ( A e. V -> ( x e. ( Clsd ` ~P A ) <-> x C_ A ) )
11		velpw 3681	. . 3 |- ( x e. ~P A <-> x C_ A )
12	10, 11	bitr4di 198	. 2 |- ( A e. V -> ( x e. ( Clsd ` ~P A ) <-> x e. ~P A ) )
13	12	eqrdv 2232	1 |- ( A e. V -> ( Clsd ` ~P A ) = ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   \ cdif 3211   C_ wss 3214   ~Pcpw 3674   U.cuni 3919   ` cfv 5357   Topctop 14988   Clsdccld 15083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-top 14989  df-cld 15086

This theorem is referenced by:  sn0cld  15128