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Theorem discld 23097
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 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)

Proof of Theorem discld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 difss 4136 . . . . 5 (𝐴𝑥) ⊆ 𝐴
2 elpw2g 5333 . . . . 5 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
31, 2mpbiri 258 . . . 4 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
4 distop 23002 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
5 unipw 5455 . . . . . . 7 𝒫 𝐴 = 𝐴
65eqcomi 2746 . . . . . 6 𝐴 = 𝒫 𝐴
76iscld 23035 . . . . 5 (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
84, 7syl 17 . . . 4 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
93, 8mpbiran2d 708 . . 3 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥𝐴))
10 velpw 4605 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
119, 10bitr4di 289 . 2 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
1211eqrdv 2735 1 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  wss 3951  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by:  sn0cld  23098
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