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Theorem dishaus 21396
Description: A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
dishaus (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)

Proof of Theorem dishaus
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 21009 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 simplrl 786 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
32snssd 4530 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ⊆ 𝐴)
4 snex 5098 . . . . . . 7 {𝑥} ∈ V
54elpw 4357 . . . . . 6 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
63, 5sylibr 225 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ∈ 𝒫 𝐴)
7 simplrr 787 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦𝐴)
87snssd 4530 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ⊆ 𝐴)
9 snex 5098 . . . . . . 7 {𝑦} ∈ V
109elpw 4357 . . . . . 6 ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
118, 10sylibr 225 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ∈ 𝒫 𝐴)
12 vsnid 4403 . . . . . 6 𝑥 ∈ {𝑥}
1312a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥 ∈ {𝑥})
14 vsnid 4403 . . . . . 6 𝑦 ∈ {𝑦}
1514a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦 ∈ {𝑦})
16 disjsn2 4439 . . . . . 6 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
1716adantl 469 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18 eleq2 2874 . . . . . . 7 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
19 ineq1 4006 . . . . . . . 8 (𝑢 = {𝑥} → (𝑢𝑣) = ({𝑥} ∩ 𝑣))
2019eqeq1d 2808 . . . . . . 7 (𝑢 = {𝑥} → ((𝑢𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
2118, 203anbi13d 1555 . . . . . 6 (𝑢 = {𝑥} → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22 eleq2 2874 . . . . . . 7 (𝑣 = {𝑦} → (𝑦𝑣𝑦 ∈ {𝑦}))
23 ineq2 4007 . . . . . . . 8 (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
2423eqeq1d 2808 . . . . . . 7 (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
2522, 243anbi23d 1556 . . . . . 6 (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
2621, 25rspc2ev 3517 . . . . 5 (({𝑥} ∈ 𝒫 𝐴 ∧ {𝑦} ∈ 𝒫 𝐴 ∧ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
276, 11, 13, 15, 17, 26syl113anc 1494 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
2827ex 399 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
2928ralrimivva 3159 . 2 (𝐴𝑉 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
30 unipw 5108 . . . 4 𝒫 𝐴 = 𝐴
3130eqcomi 2815 . . 3 𝐴 = 𝒫 𝐴
3231ishaus 21336 . 2 (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))))
331, 29, 32sylanbrc 574 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096  wrex 3097  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  {csn 4370   cuni 4630  Topctop 20907  Hauscha 21322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-uni 4631  df-top 20908  df-haus 21329
This theorem is referenced by:  ssoninhaus  32759
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