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Theorem discld 21673
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 4084 . . . . 5 (𝐴𝑥) ⊆ 𝐴
2 elpw2g 5220 . . . . 5 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
31, 2mpbiri 261 . . . 4 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
4 distop 21579 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
5 unipw 5316 . . . . . . 7 𝒫 𝐴 = 𝐴
65eqcomi 2830 . . . . . 6 𝐴 = 𝒫 𝐴
76iscld 21611 . . . . 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 707 . . 3 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥𝐴))
10 velpw 4517 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
119, 10syl6bbr 292 . 2 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
1211eqrdv 2819 1 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cdif 3907  wss 3910  𝒫 cpw 4512   cuni 4811  cfv 6328  Topctop 21477  Clsdccld 21600
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-top 21478  df-cld 21603
This theorem is referenced by:  sn0cld  21674
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