Theorem discld 22983

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 4102	. . . . 5 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
2		elpw2g 5291	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
3	1, 2	mpbiri 258	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
4		distop 22889	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
5		unipw 5413	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
6	5	eqcomi 2739	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
7	6	iscld 22921	. . . . 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 708	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ⊆ 𝐴))
10		velpw 4571	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
11	9, 10	bitr4di 289	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
12	11	eqrdv 2728	1 ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ‘cfv 6514  Topctop 22787  Clsdccld 22910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-top 22788  df-cld 22913

This theorem is referenced by:  sn0cld  22984