Theorem discld 23054

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 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Proof of Theorem discld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 4076	. . . . 5 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
2		elpw2g 5274	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
3	1, 2	mpbiri 258	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
4		distop 22960	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
5		unipw 5402	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
6	5	eqcomi 2745	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
7	6	iscld 22992	. . . . 5 ⊢ (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)))
8	4, 7	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)))
9	3, 8	mpbiran2d 709	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ⊆ 𝐴))
10		velpw 4546	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
11	9, 10	bitr4di 289	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
12	11	eqrdv 2734	1 ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  Clsdccld 22981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-top 22859  df-cld 22984

This theorem is referenced by:  sn0cld  23055