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Theorem sn0topon 22851
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 4810 . 2 𝒫 ∅ = {∅}
2 0ex 5300 . . 3 ∅ ∈ V
3 distopon 22850 . . 3 (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅))
42, 3ax-mp 5 . 2 𝒫 ∅ ∈ (TopOn‘∅)
51, 4eqeltrri 2824 1 {∅} ∈ (TopOn‘∅)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3468  c0 4317  𝒫 cpw 4597  {csn 4623  cfv 6536  TopOnctopon 22762
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-top 22746  df-topon 22763
This theorem is referenced by:  sn0top  22852  0cnf  45147
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