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Mirrors > Home > MPE Home > Th. List > sn0topon | Structured version Visualization version GIF version |
Description: The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
sn0topon | ⊢ {∅} ∈ (TopOn‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw0 4745 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 5231 | . . 3 ⊢ ∅ ∈ V | |
3 | distopon 22147 | . . 3 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝒫 ∅ ∈ (TopOn‘∅) |
5 | 1, 4 | eqeltrri 2836 | 1 ⊢ {∅} ∈ (TopOn‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3432 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ‘cfv 6433 TopOnctopon 22059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-top 22043 df-topon 22060 |
This theorem is referenced by: sn0top 22149 0cnf 43418 |
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