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| Description: The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.) | 
| Ref | Expression | 
|---|---|
| sn0cld | ⊢ (Clsd‘{∅}) = {∅} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 2 | discld 23097 | . . 3 ⊢ (∅ ∈ V → (Clsd‘𝒫 ∅) = 𝒫 ∅) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ (Clsd‘𝒫 ∅) = 𝒫 ∅ | 
| 4 | pw0 4812 | . . 3 ⊢ 𝒫 ∅ = {∅} | |
| 5 | 4 | fveq2i 6909 | . 2 ⊢ (Clsd‘𝒫 ∅) = (Clsd‘{∅}) | 
| 6 | 3, 5, 4 | 3eqtr3i 2773 | 1 ⊢ (Clsd‘{∅}) = {∅} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ‘cfv 6561 Clsdccld 23024 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-top 22900 df-cld 23027 | 
| This theorem is referenced by: (None) | 
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