Theorem discld 14810

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 14759	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		unipw 4303	. . . . . . 7 |- U. ~P A = A
3	2	eqcomi 2233	. . . . . 6 |- A = U. ~P A
4	3	iscld 14777	. . . . 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 3330	. . . . . 6 |- ( A \ x ) C_ A
7		elpw2g 4240	. . . . . 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 3656	. . 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 2227	1 |- ( A e. V -> ( Clsd ` ~P A ) = ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   \ cdif 3194   C_ wss 3197   ~Pcpw 3649   U.cuni 3888   ` cfv 5318   Topctop 14671   Clsdccld 14766

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-top 14672  df-cld 14769

This theorem is referenced by:  sn0cld  14811