Theorem discld 14859

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 14808	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		unipw 4309	. . . . . . 7 |- U. ~P A = A
3	2	eqcomi 2235	. . . . . 6 |- A = U. ~P A
4	3	iscld 14826	. . . . 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 3333	. . . . . 6 |- ( A \ x ) C_ A
7		elpw2g 4246	. . . . . 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 3659	. . 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 2229	1 |- ( A e. V -> ( Clsd ` ~P A ) = ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   \ cdif 3197   C_ wss 3200   ~Pcpw 3652   U.cuni 3893   ` cfv 5326   Topctop 14720   Clsdccld 14815

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-top 14721  df-cld 14818

This theorem is referenced by:  sn0cld  14860