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Theorem discld 14315
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 14264 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 4247 . . . . . . 7 𝒫 𝐴 = 𝐴
32eqcomi 2197 . . . . . 6 𝐴 = 𝒫 𝐴
43iscld 14282 . . . . 5 (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
51, 4syl 14 . . . 4 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
6 difss 3286 . . . . . 6 (𝐴𝑥) ⊆ 𝐴
7 elpw2g 4186 . . . . . 6 (𝐴𝑉 → ((𝐴𝑥) ∈ 𝒫 𝐴 ↔ (𝐴𝑥) ⊆ 𝐴))
86, 7mpbiri 168 . . . . 5 (𝐴𝑉 → (𝐴𝑥) ∈ 𝒫 𝐴)
98biantrud 304 . . . 4 (𝐴𝑉 → (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝐴𝑥) ∈ 𝒫 𝐴)))
105, 9bitr4d 191 . . 3 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥𝐴))
11 velpw 3609 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1210, 11bitr4di 198 . 2 (𝐴𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴))
1312eqrdv 2191 1 (𝐴𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  cdif 3151  wss 3154  𝒫 cpw 3602   cuni 3836  cfv 5255  Topctop 14176  Clsdccld 14271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-top 14177  df-cld 14274
This theorem is referenced by:  sn0cld  14316
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