MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dishaus Structured version   Visualization version   GIF version

Theorem dishaus 21234
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 20847 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 simplrl 817 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
32snssd 4372 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ⊆ 𝐴)
4 snex 4938 . . . . . . 7 {𝑥} ∈ V
54elpw 4197 . . . . . 6 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
63, 5sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ∈ 𝒫 𝐴)
7 simplrr 818 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦𝐴)
87snssd 4372 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ⊆ 𝐴)
9 snex 4938 . . . . . . 7 {𝑦} ∈ V
109elpw 4197 . . . . . 6 ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
118, 10sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ∈ 𝒫 𝐴)
12 vsnid 4242 . . . . . 6 𝑥 ∈ {𝑥}
1312a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥 ∈ {𝑥})
14 vsnid 4242 . . . . . 6 𝑦 ∈ {𝑦}
1514a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦 ∈ {𝑦})
16 disjsn2 4279 . . . . . 6 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
1716adantl 481 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18 eleq2 2719 . . . . . . 7 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
19 ineq1 3840 . . . . . . . 8 (𝑢 = {𝑥} → (𝑢𝑣) = ({𝑥} ∩ 𝑣))
2019eqeq1d 2653 . . . . . . 7 (𝑢 = {𝑥} → ((𝑢𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
2118, 203anbi13d 1441 . . . . . 6 (𝑢 = {𝑥} → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22 eleq2 2719 . . . . . . 7 (𝑣 = {𝑦} → (𝑦𝑣𝑦 ∈ {𝑦}))
23 ineq2 3841 . . . . . . . 8 (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
2423eqeq1d 2653 . . . . . . 7 (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
2522, 243anbi23d 1442 . . . . . 6 (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
2621, 25rspc2ev 3355 . . . . 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 1378 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
2827ex 449 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
2928ralrimivva 3000 . 2 (𝐴𝑉 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
30 unipw 4948 . . . 4 𝒫 𝐴 = 𝐴
3130eqcomi 2660 . . 3 𝐴 = 𝒫 𝐴
3231ishaus 21174 . 2 (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))))
331, 29, 32sylanbrc 699 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  Topctop 20746  Hauscha 21160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469  df-top 20747  df-haus 21167
This theorem is referenced by:  ssoninhaus  32572
  Copyright terms: Public domain W3C validator