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Theorem distopon 20892
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 20890 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 4991 . . . 4 𝒫 𝐴 = 𝐴
32eqcomi 2701 . . 3 𝐴 = 𝒫 𝐴
43a1i 11 . 2 (𝐴𝑉𝐴 = 𝒫 𝐴)
5 istopon 20808 . 2 (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = 𝒫 𝐴))
61, 4, 5sylanbrc 701 1 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1564  wcel 2071  𝒫 cpw 4234   cuni 4512  cfv 5969  Topctop 20789  TopOnctopon 20806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-iota 5932  df-fun 5971  df-fv 5977  df-top 20790  df-topon 20807
This theorem is referenced by:  sn0topon  20893  toponmre  20988  cndis  21186  txdis1cn  21529  xkofvcn  21578  distgp  21993  symgtgp  21995  cnfdmsn  40483
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