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Theorem discld 21301
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 21207 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 5150 . . . . . . 7 𝒫 𝐴 = 𝐴
32eqcomi 2787 . . . . . 6 𝐴 = 𝒫 𝐴
43iscld 21239 . . . . 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 3960 . . . . . 6 (𝐴𝑥) ⊆ 𝐴
7 elpw2g 5061 . . . . . 6 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
86, 7mpbiri 250 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
98biantrud 527 . . . 4 (𝐴𝑉 → (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
105, 9bitr4d 274 . . 3 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥𝐴))
11 selpw 4386 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1210, 11syl6bbr 281 . 2 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
1312eqrdv 2776 1 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  cdif 3789  wss 3792  𝒫 cpw 4379   cuni 4671  cfv 6135  Topctop 21105  Clsdccld 21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-top 21106  df-cld 21231
This theorem is referenced by:  sn0cld  21302
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