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Theorem sn0topon 22921
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 4820 . 2 𝒫 ∅ = {∅}
2 0ex 5311 . . 3 ∅ ∈ V
3 distopon 22920 . . 3 (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅))
42, 3ax-mp 5 . 2 𝒫 ∅ ∈ (TopOn‘∅)
51, 4eqeltrri 2826 1 {∅} ∈ (TopOn‘∅)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3473  c0 4326  𝒫 cpw 4606  {csn 4632  cfv 6553  TopOnctopon 22832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-top 22816  df-topon 22833
This theorem is referenced by:  sn0top  22922  0cnf  45294
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