Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > discld | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
discld | ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4108 | . . . . 5 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 | |
2 | elpw2g 5240 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑥) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑥) ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 260 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴) |
4 | distop 21597 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | |
5 | unipw 5335 | . . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
6 | 5 | eqcomi 2830 | . . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
7 | 6 | iscld 21629 | . . . . 5 ⊢ (𝒫 𝐴 ∈ Top → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴))) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝒫 𝐴))) |
9 | 3, 8 | mpbiran2d 706 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ⊆ 𝐴)) |
10 | velpw 4547 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
11 | 9, 10 | syl6bbr 291 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Clsd‘𝒫 𝐴) ↔ 𝑥 ∈ 𝒫 𝐴)) |
12 | 11 | eqrdv 2819 | 1 ⊢ (𝐴 ∈ 𝑉 → (Clsd‘𝒫 𝐴) = 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 ‘cfv 6350 Topctop 21495 Clsdccld 21618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-top 21496 df-cld 21621 |
This theorem is referenced by: sn0cld 21692 |
Copyright terms: Public domain | W3C validator |