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Theorem sn0topon 22986
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 4811 . 2 𝒫 ∅ = {∅}
2 0ex 5302 . . 3 ∅ ∈ V
3 distopon 22985 . . 3 (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅))
42, 3ax-mp 5 . 2 𝒫 ∅ ∈ (TopOn‘∅)
51, 4eqeltrri 2823 1 {∅} ∈ (TopOn‘∅)
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  Vcvv 3462  c0 4322  𝒫 cpw 4597  {csn 4623  cfv 6543  TopOnctopon 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22881  df-topon 22898
This theorem is referenced by:  sn0top  22987  0cnf  45531
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