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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 4748 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 5214 | . . 3 ⊢ ∅ ∈ V | |
3 | distopon 21608 | . . 3 ⊢ (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝒫 ∅ ∈ (TopOn‘∅) |
5 | 1, 4 | eqeltrri 2913 | 1 ⊢ {∅} ∈ (TopOn‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3497 ∅c0 4294 𝒫 cpw 4542 {csn 4570 ‘cfv 6358 TopOnctopon 21521 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-top 21505 df-topon 21522 |
This theorem is referenced by: sn0top 21610 0cnf 42166 |
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