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Mirrors > Home > MPE Home > Th. List > sn0cld | Structured version Visualization version GIF version |
Description: The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.) |
Ref | Expression |
---|---|
sn0cld | ⊢ (Clsd‘{∅}) = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5312 | . . 3 ⊢ ∅ ∈ V | |
2 | discld 23112 | . . 3 ⊢ (∅ ∈ V → (Clsd‘𝒫 ∅) = 𝒫 ∅) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (Clsd‘𝒫 ∅) = 𝒫 ∅ |
4 | pw0 4816 | . . 3 ⊢ 𝒫 ∅ = {∅} | |
5 | 4 | fveq2i 6909 | . 2 ⊢ (Clsd‘𝒫 ∅) = (Clsd‘{∅}) |
6 | 3, 5, 4 | 3eqtr3i 2770 | 1 ⊢ (Clsd‘{∅}) = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 𝒫 cpw 4604 {csn 4630 ‘cfv 6562 Clsdccld 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-top 22915 df-cld 23042 |
This theorem is referenced by: (None) |
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