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Theorem distopon 23020
Description: The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
distopon (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))

Proof of Theorem distopon
StepHypRef Expression
1 distop 23018 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 5461 . . 3 𝒫 𝐴 = 𝐴
32eqcomi 2744 . 2 𝐴 = 𝒫 𝐴
4 istopon 22934 . 2 (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = 𝒫 𝐴))
51, 3, 4sylanblrc 590 1 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  𝒫 cpw 4605   cuni 4912  cfv 6563  Topctop 22915  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:  sn0topon  23021  toponmre  23117  cndis  23315  txdis1cn  23659  xkofvcn  23708  distgp  24123  efmndtmd  24125  symgtgp  24130  cnfdmsn  45838
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