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Theorem distopon 23005
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 23003 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 5454 . . 3 𝒫 𝐴 = 𝐴
32eqcomi 2745 . 2 𝐴 = 𝒫 𝐴
4 istopon 22919 . 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 1539  wcel 2107  𝒫 cpw 4599   cuni 4906  cfv 6560  Topctop 22900  TopOnctopon 22917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-topon 22918
This theorem is referenced by:  sn0topon  23006  toponmre  23102  cndis  23300  txdis1cn  23644  xkofvcn  23693  distgp  24108  efmndtmd  24110  symgtgp  24115  cnfdmsn  45902
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