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Theorem distop 20610
 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 4394 . . . . . 6 (𝑥 ⊆ 𝒫 𝐴 𝑥 𝒫 𝐴)
2 unipw 4845 . . . . . 6 𝒫 𝐴 = 𝐴
31, 2syl6sseq 3614 . . . . 5 (𝑥 ⊆ 𝒫 𝐴 𝑥𝐴)
4 vuniex 6852 . . . . . 6 𝑥 ∈ V
54elpw 4114 . . . . 5 ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴)
63, 5sylibr 223 . . . 4 (𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
76ax-gen 1713 . . 3 𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴)
87a1i 11 . 2 (𝐴𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 𝑥 ∈ 𝒫 𝐴))
9 selpw 4115 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
10 selpw 4115 . . . . . . . 8 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
11 ssinss1 3803 . . . . . . . . . 10 (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴)
1211a1i 11 . . . . . . . . 9 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ⊆ 𝐴))
13 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
1413inex2 4728 . . . . . . . . . 10 (𝑥𝑦) ∈ V
1514elpw 4114 . . . . . . . . 9 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
1612, 15syl6ibr 241 . . . . . . . 8 (𝑦𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1710, 16sylbi 206 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴 → (𝑥𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
1817com12 32 . . . . . 6 (𝑥𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
199, 18sylbi 206 . . . . 5 (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴))
2019ralrimiv 2948 . . . 4 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
2120rgen 2906 . . 3 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
2221a1i 11 . 2 (𝐴𝑉 → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴)
23 pwexg 4776 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
24 istopg 20525 . . 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 959 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  Topctop 20517 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-top 20521 This theorem is referenced by:  distopon  20611  distps  20629  discld  20703  restdis  20792  dishaus  20996  discmp  21011  dis2ndc  21073  dislly  21110  dis1stc  21112  dissnlocfin  21142  locfindis  21143  txdis  21245  xkopt  21268  xkofvcn  21297  symgtgp  21715  dispcmp  29254  bj-topnex  32247
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