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Theorem sn0topon 23021
Description: The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
sn0topon {∅} ∈ (TopOn‘∅)

Proof of Theorem sn0topon
StepHypRef Expression
1 pw0 4817 . 2 𝒫 ∅ = {∅}
2 0ex 5313 . . 3 ∅ ∈ V
3 distopon 23020 . . 3 (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅))
42, 3ax-mp 5 . 2 𝒫 ∅ ∈ (TopOn‘∅)
51, 4eqeltrri 2836 1 {∅} ∈ (TopOn‘∅)
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  c0 4339  𝒫 cpw 4605  {csn 4631  cfv 6563  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933
This theorem is referenced by:  sn0top  23022  0cnf  45833
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