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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 4532 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 4985 | . . 3 ⊢ ∅ ∈ V | |
3 | distopon 21129 | . . 3 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝒫 ∅ ∈ (TopOn‘∅) |
5 | 1, 4 | eqeltrri 2876 | 1 ⊢ {∅} ∈ (TopOn‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 Vcvv 3386 ∅c0 4116 𝒫 cpw 4350 {csn 4369 ‘cfv 6102 TopOnctopon 21042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-iota 6065 df-fun 6104 df-fv 6110 df-top 21026 df-topon 21043 |
This theorem is referenced by: sn0top 21131 0cnf 40829 |
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