Theorem discld 23064

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 4077	. . . . 5 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴
2		elpw2g 5270	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴))
3	1, 2	mpbiri 258	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴)
4		distop 22970	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
5		unipw 5397	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
6	5	eqcomi 2746	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
7	6	iscld 23002	. . . . 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 4547	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
11	9, 10	bitr4di 289	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
12	11	eqrdv 2735	1 ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  Topctop 22868  Clsdccld 22991

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 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-top 22869  df-cld 22994

This theorem is referenced by:  sn0cld  23065