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Theorem dishaus 21906
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 21519 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 simplrl 773 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
32snssd 4741 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ⊆ 𝐴)
4 snex 5328 . . . . . . 7 {𝑥} ∈ V
54elpw 4549 . . . . . 6 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
63, 5sylibr 235 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ∈ 𝒫 𝐴)
7 simplrr 774 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦𝐴)
87snssd 4741 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ⊆ 𝐴)
9 snex 5328 . . . . . . 7 {𝑦} ∈ V
109elpw 4549 . . . . . 6 ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
118, 10sylibr 235 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ∈ 𝒫 𝐴)
12 vsnid 4599 . . . . . 6 𝑥 ∈ {𝑥}
1312a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥 ∈ {𝑥})
14 vsnid 4599 . . . . . 6 𝑦 ∈ {𝑦}
1514a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦 ∈ {𝑦})
16 disjsn2 4647 . . . . . 6 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
1716adantl 482 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18 eleq2 2906 . . . . . . 7 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
19 ineq1 4185 . . . . . . . 8 (𝑢 = {𝑥} → (𝑢𝑣) = ({𝑥} ∩ 𝑣))
2019eqeq1d 2828 . . . . . . 7 (𝑢 = {𝑥} → ((𝑢𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
2118, 203anbi13d 1431 . . . . . 6 (𝑢 = {𝑥} → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22 eleq2 2906 . . . . . . 7 (𝑣 = {𝑦} → (𝑦𝑣𝑦 ∈ {𝑦}))
23 ineq2 4187 . . . . . . . 8 (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
2423eqeq1d 2828 . . . . . . 7 (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
2522, 243anbi23d 1432 . . . . . 6 (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
2621, 25rspc2ev 3639 . . . . 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 1376 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
2827ex 413 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
2928ralrimivva 3196 . 2 (𝐴𝑉 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
30 unipw 5339 . . . 4 𝒫 𝐴 = 𝐴
3130eqcomi 2835 . . 3 𝐴 = 𝒫 𝐴
3231ishaus 21846 . 2 (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))))
331, 29, 32sylanbrc 583 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   cuni 4837  Topctop 21417  Hauscha 21832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-pw 4544  df-sn 4565  df-pr 4567  df-uni 4838  df-top 21418  df-haus 21839
This theorem is referenced by:  ssoninhaus  33680
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