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Theorem distop 20513
Description: The discrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
distop (𝐴𝑉 → 𝒫 𝐴 ∈ Top)

Proof of Theorem distop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniss 4292 . . . . . 6 (𝑥 ⊆ 𝒫 𝐴 𝑥 𝒫 𝐴)
2 unipw 4743 . . . . . 6 𝒫 𝐴 = 𝐴
31, 2syl6sseq 3518 . . . . 5 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
4 vuniex 6728 . . . . . 6 𝑥 ∈ V
54elpw 4017 . . . . 5 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
63, 5sylibr 222 . . . 4 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
76ax-gen 1700 . . 3 𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
87a1i 11 . 2 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴))
9 selpw 4018 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
10 selpw 4018 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
11 ssinss1 3706 . . . . . . . . . 10 (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴)
1211a1i 11 . . . . . . . . 9 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴))
13 vex 3080 . . . . . . . . . . 11 𝑦 ∈ V
1413inex2 4627 . . . . . . . . . 10 (𝑥𝑦) ∈ V
1514elpw 4017 . . . . . . . . 9 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
1612, 15syl6ibr 240 . . . . . . . 8 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1710, 16sylbi 205 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1817com12 32 . . . . . 6 (𝑥𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
199, 18sylbi 205 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
2019ralrimiv 2852 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
2120rgen 2810 . . 3 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
2221a1i 11 . 2 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
23 pwexg 4675 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
24 istopg 20428 . . 3 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)))
2523, 24syl 17 . 2 (𝐴𝑉 → (𝒫 𝐴 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)))
268, 22, 25mpbir2and 958 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1938  wral 2800  Vcvv 3077  cin 3443  wss 3444  𝒫 cpw 4011   cuni 4270  Topctop 20420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-pw 4013  df-sn 4029  df-pr 4031  df-uni 4271  df-top 20424
This theorem is referenced by:  distopon  20514  distps  20532  discld  20606  restdis  20695  dishaus  20899  discmp  20914  dis2ndc  20976  dislly  21013  dis1stc  21015  dissnlocfin  21045  locfindis  21046  txdis  21148  xkopt  21171  xkofvcn  21200  symgtgp  21618  dispcmp  29050  bj-topnex  32078
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