Theorem discld 15001

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 14950	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		unipw 4333	. . . . . . 7 |- U. ~P A = A
3	2	eqcomi 2236	. . . . . 6 |- A = U. ~P A
4	3	iscld 14968	. . . . 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 3345	. . . . . 6 |- ( A \ x ) C_ A
7		elpw2g 4268	. . . . . 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 3676	. . 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 2230	1 |- ( A e. V -> ( Clsd ` ~P A ) = ~P A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   \ cdif 3208   C_ wss 3211   ~Pcpw 3669   U.cuni 3914   ` cfv 5352   Topctop 14862   Clsdccld 14957

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-top 14863  df-cld 14960

This theorem is referenced by:  sn0cld  15002