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Theorem discld 20703
 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 distop 20610 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 4845 . . . . . . 7 𝒫 𝐴 = 𝐴
32eqcomi 2619 . . . . . 6 𝐴 = 𝒫 𝐴
43iscld 20641 . . . . 5 (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
51, 4syl 17 . . . 4 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
6 difss 3699 . . . . . 6 (𝐴𝑥) ⊆ 𝐴
7 elpw2g 4754 . . . . . 6 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
86, 7mpbiri 247 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
98biantrud 527 . . . 4 (𝐴𝑉 → (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
105, 9bitr4d 270 . . 3 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥𝐴))
11 selpw 4115 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1210, 11syl6bbr 277 . 2 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
1312eqrdv 2608 1 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-cld 20633 This theorem is referenced by:  sn0cld  20704
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