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Theorem distop 22989
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 4921 . . . . . 6 (𝑥 ⊆ 𝒫 𝐴 𝑥 𝒫 𝐴)
2 unipw 5456 . . . . . 6 𝒫 𝐴 = 𝐴
31, 2sseqtrdi 4030 . . . . 5 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
4 vuniex 7750 . . . . . 6 𝑥 ∈ V
54elpw 4611 . . . . 5 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
63, 5sylibr 233 . . . 4 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
76ax-gen 1790 . . 3 𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
87a1i 11 . 2 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴))
9 velpw 4612 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
10 velpw 4612 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
11 ssinss1 4239 . . . . . . . . . 10 (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴)
1211a1i 11 . . . . . . . . 9 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴))
13 vex 3466 . . . . . . . . . . 11 𝑦 ∈ V
1413inex2 5323 . . . . . . . . . 10 (𝑥𝑦) ∈ V
1514elpw 4611 . . . . . . . . 9 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
1612, 15imbitrrdi 251 . . . . . . . 8 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1710, 16sylbi 216 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1817com12 32 . . . . . 6 (𝑥𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
199, 18sylbi 216 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
2019ralrimiv 3135 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
2120rgen 3053 . . 3 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
2221a1i 11 . 2 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
23 pwexg 5382 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
24 istopg 22888 . . 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 711 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1532  wcel 2099  wral 3051  Vcvv 3462  cin 3946  wss 3947  𝒫 cpw 4607   cuni 4913  Topctop 22886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-un 3952  df-in 3954  df-ss 3964  df-pw 4609  df-sn 4634  df-pr 4636  df-uni 4914  df-top 22887
This theorem is referenced by:  topnex  22990  distopon  22991  distps  23010  discld  23084  restdis  23173  dishaus  23377  discmp  23393  dis2ndc  23455  dislly  23492  dis1stc  23494  dissnlocfin  23524  locfindis  23525  txdis  23627  xkopt  23650  xkofvcn  23679  efmndtmd  24096  symgtgp  24101  dispcmp  33674
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