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Theorem distopon 21081
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 21079 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 5074 . . . 4 𝒫 𝐴 = 𝐴
32eqcomi 2774 . . 3 𝐴 = 𝒫 𝐴
43a1i 11 . 2 (𝐴𝑉𝐴 = 𝒫 𝐴)
5 istopon 20996 . 2 (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = 𝒫 𝐴))
61, 4, 5sylanbrc 578 1 (𝐴𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  𝒫 cpw 4315   cuni 4594  cfv 6068  Topctop 20977  TopOnctopon 20994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-top 20978  df-topon 20995
This theorem is referenced by:  sn0topon  21082  toponmre  21177  cndis  21375  txdis1cn  21718  xkofvcn  21767  distgp  22182  symgtgp  22184  cnfdmsn  40665
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