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Theorem dishaus 21407
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 21020 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 simplrl 762 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
32snssd 4475 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ⊆ 𝐴)
4 snex 5036 . . . . . . 7 {𝑥} ∈ V
54elpw 4303 . . . . . 6 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
63, 5sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ∈ 𝒫 𝐴)
7 simplrr 763 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦𝐴)
87snssd 4475 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ⊆ 𝐴)
9 snex 5036 . . . . . . 7 {𝑦} ∈ V
109elpw 4303 . . . . . 6 ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
118, 10sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ∈ 𝒫 𝐴)
12 vsnid 4348 . . . . . 6 𝑥 ∈ {𝑥}
1312a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥 ∈ {𝑥})
14 vsnid 4348 . . . . . 6 𝑦 ∈ {𝑦}
1514a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦 ∈ {𝑦})
16 disjsn2 4384 . . . . . 6 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
1716adantl 467 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18 eleq2 2839 . . . . . . 7 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
19 ineq1 3958 . . . . . . . 8 (𝑢 = {𝑥} → (𝑢𝑣) = ({𝑥} ∩ 𝑣))
2019eqeq1d 2773 . . . . . . 7 (𝑢 = {𝑥} → ((𝑢𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
2118, 203anbi13d 1549 . . . . . 6 (𝑢 = {𝑥} → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22 eleq2 2839 . . . . . . 7 (𝑣 = {𝑦} → (𝑦𝑣𝑦 ∈ {𝑦}))
23 ineq2 3959 . . . . . . . 8 (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
2423eqeq1d 2773 . . . . . . 7 (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
2522, 243anbi23d 1550 . . . . . 6 (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
2621, 25rspc2ev 3474 . . . . 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 1488 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
2827ex 397 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
2928ralrimivva 3120 . 2 (𝐴𝑉 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
30 unipw 5046 . . . 4 𝒫 𝐴 = 𝐴
3130eqcomi 2780 . . 3 𝐴 = 𝒫 𝐴
3231ishaus 21347 . 2 (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))))
331, 29, 32sylanbrc 572 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   cuni 4574  Topctop 20918  Hauscha 21333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-uni 4575  df-top 20919  df-haus 21340
This theorem is referenced by:  ssoninhaus  32784
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