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Mirrors > Home > ILE Home > Th. List > sn0topon | 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 3714 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 4103 | . . 3 ⊢ ∅ ∈ V | |
3 | distopon 12634 | . . 3 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝒫 ∅ ∈ (TopOn‘∅) |
5 | 1, 4 | eqeltrri 2238 | 1 ⊢ {∅} ∈ (TopOn‘∅) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Vcvv 2721 ∅c0 3404 𝒫 cpw 3553 {csn 3570 ‘cfv 5182 TopOnctopon 12555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-top 12543 df-topon 12556 |
This theorem is referenced by: sn0top 12636 |
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