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Theorem distopon 21024
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 21022 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 unipw 5068 . . . 4 𝒫 𝐴 = 𝐴
32eqcomi 2770 . . 3 𝐴 = 𝒫 𝐴
43a1i 11 . 2 (𝐴𝑉𝐴 = 𝒫 𝐴)
5 istopon 20940 . 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 1632  wcel 2140  𝒫 cpw 4303   cuni 4589  cfv 6050  Topctop 20921  TopOnctopon 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-top 20922  df-topon 20939
This theorem is referenced by:  sn0topon  21025  toponmre  21120  cndis  21318  txdis1cn  21661  xkofvcn  21710  distgp  22125  symgtgp  22127  cnfdmsn  40617
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