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

Theorem dishaus 21091
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 20705 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 simplrl 799 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥𝐴)
32snssd 4314 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ⊆ 𝐴)
4 snex 4874 . . . . . . 7 {𝑥} ∈ V
54elpw 4141 . . . . . 6 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
63, 5sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑥} ∈ 𝒫 𝐴)
7 simplrr 800 . . . . . . 7 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦𝐴)
87snssd 4314 . . . . . 6 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ⊆ 𝐴)
9 snex 4874 . . . . . . 7 {𝑦} ∈ V
109elpw 4141 . . . . . 6 ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
118, 10sylibr 224 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → {𝑦} ∈ 𝒫 𝐴)
12 vsnid 4185 . . . . . 6 𝑥 ∈ {𝑥}
1312a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑥 ∈ {𝑥})
14 vsnid 4185 . . . . . 6 𝑦 ∈ {𝑦}
1514a1i 11 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → 𝑦 ∈ {𝑦})
16 disjsn2 4222 . . . . . 6 (𝑥𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
1716adantl 482 . . . . 5 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18 eleq2 2693 . . . . . . 7 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
19 ineq1 3790 . . . . . . . 8 (𝑢 = {𝑥} → (𝑢𝑣) = ({𝑥} ∩ 𝑣))
2019eqeq1d 2628 . . . . . . 7 (𝑢 = {𝑥} → ((𝑢𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
2118, 203anbi13d 1398 . . . . . 6 (𝑢 = {𝑥} → ((𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22 eleq2 2693 . . . . . . 7 (𝑣 = {𝑦} → (𝑦𝑣𝑦 ∈ {𝑦}))
23 ineq2 3791 . . . . . . . 8 (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
2423eqeq1d 2628 . . . . . . 7 (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
2522, 243anbi23d 1399 . . . . . 6 (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
2621, 25rspc2ev 3313 . . . . 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 1335 . . . 4 (((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) ∧ 𝑥𝑦) → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))
2827ex 450 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
2928ralrimivva 2970 . 2 (𝐴𝑉 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅)))
30 unipw 4884 . . . 4 𝒫 𝐴 = 𝐴
3130eqcomi 2635 . . 3 𝐴 = 𝒫 𝐴
3231ishaus 21031 . 2 (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑢 ∈ 𝒫 𝐴𝑣 ∈ 𝒫 𝐴(𝑥𝑢𝑦𝑣 ∧ (𝑢𝑣) = ∅))))
331, 29, 32sylanbrc 697 1 (𝐴𝑉 → 𝒫 𝐴 ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cin 3559  wss 3560  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  Topctop 20612  Hauscha 21017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-top 20616  df-haus 21024
This theorem is referenced by:  ssoninhaus  32081
  Copyright terms: Public domain W3C validator