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Theorem sn0topon 21130
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 4532 . 2 𝒫 ∅ = {∅}
2 0ex 4985 . . 3 ∅ ∈ V
3 distopon 21129 . . 3 (∅ ∈ V → 𝒫 ∅ ∈ (TopOn‘∅))
42, 3ax-mp 5 . 2 𝒫 ∅ ∈ (TopOn‘∅)
51, 4eqeltrri 2876 1 {∅} ∈ (TopOn‘∅)
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  Vcvv 3386  c0 4116  𝒫 cpw 4350  {csn 4369  cfv 6102  TopOnctopon 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-top 21026  df-topon 21043
This theorem is referenced by:  sn0top  21131  0cnf  40829
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