Theorem discld 20893

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		distop 20799	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		unipw 4918	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	2	eqcomi 2631	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
4	3	iscld 20831	. . . . 5 ⊢ (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)))
5	1, 4	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)))
6		difss 3737	. . . . . 6 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
7		elpw2g 4827	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
8	6, 7	mpbiri 248	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
9	8	biantrud 528	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ⊆ 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)))
10	5, 9	bitr4d 271	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ⊆ 𝐴))
11		selpw 4165	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
12	10, 11	syl6bbr 278	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
13	12	eqrdv 2620	1 ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-cld 20823

This theorem is referenced by:  sn0cld  20894