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Theorem distop 21604
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 4811 . . . . . 6 (𝑥 ⊆ 𝒫 𝐴 𝑥 𝒫 𝐴)
2 unipw 5311 . . . . . 6 𝒫 𝐴 = 𝐴
31, 2sseqtrdi 3968 . . . . 5 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
4 vuniex 7449 . . . . . 6 𝑥 ∈ V
54elpw 4504 . . . . 5 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
63, 5sylibr 237 . . . 4 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
76ax-gen 1797 . . 3 𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
87a1i 11 . 2 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴))
9 velpw 4505 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
10 velpw 4505 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
11 ssinss1 4167 . . . . . . . . . 10 (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴)
1211a1i 11 . . . . . . . . 9 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴))
13 vex 3447 . . . . . . . . . . 11 𝑦 ∈ V
1413inex2 5189 . . . . . . . . . 10 (𝑥𝑦) ∈ V
1514elpw 4504 . . . . . . . . 9 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
1612, 15syl6ibr 255 . . . . . . . 8 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1710, 16sylbi 220 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1817com12 32 . . . . . 6 (𝑥𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
199, 18sylbi 220 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
2019ralrimiv 3151 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
2120rgen 3119 . . 3 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
2221a1i 11 . 2 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
23 pwexg 5247 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
24 istopg 21504 . . 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 712 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wcel 2112  wral 3109  Vcvv 3444  cin 3883  wss 3884  𝒫 cpw 4500   cuni 4803  Topctop 21502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4804  df-top 21503
This theorem is referenced by:  topnex  21605  distopon  21606  distps  21624  discld  21698  restdis  21787  dishaus  21991  discmp  22007  dis2ndc  22069  dislly  22106  dis1stc  22108  dissnlocfin  22138  locfindis  22139  txdis  22241  xkopt  22264  xkofvcn  22293  efmndtmd  22710  symgtgp  22715  dispcmp  31216
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