Theorem dishaus 20996

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	⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus)

Proof of Theorem dishaus

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 20610	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		simplrl 796	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝐴)
3	2	snssd 4281	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑥} ⊆ 𝐴)
4		snex 4835	. . . . . . 7 ⊢ {𝑥} ∈ V
5	4	elpw 4114	. . . . . 6 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
6	3, 5	sylibr 223	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑥} ∈ 𝒫 𝐴)
7		simplrr 797	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝐴)
8	7	snssd 4281	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑦} ⊆ 𝐴)
9		snex 4835	. . . . . . 7 ⊢ {𝑦} ∈ V
10	9	elpw 4114	. . . . . 6 ⊢ ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
11	8, 10	sylibr 223	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑦} ∈ 𝒫 𝐴)
12		vsnid 4156	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
13	12	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ {𝑥})
14		vsnid 4156	. . . . . 6 ⊢ 𝑦 ∈ {𝑦}
15	14	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ {𝑦})
16		disjsn2 4193	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
17	16	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18		eleq2 2677	. . . . . . 7 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
19		ineq1 3769	. . . . . . . 8 ⊢ (𝑢 = {𝑥} → (𝑢 ∩ 𝑣) = ({𝑥} ∩ 𝑣))
20	19	eqeq1d 2612	. . . . . . 7 ⊢ (𝑢 = {𝑥} → ((𝑢 ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
21	18, 20	3anbi13d 1393	. . . . . 6 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22		eleq2 2677	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ {𝑦}))
23		ineq2 3770	. . . . . . . 8 ⊢ (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
24	23	eqeq1d 2612	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
25	22, 24	3anbi23d 1394	. . . . . 6 ⊢ (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
26	21, 25	rspc2ev 3295	. . . . 5 ⊢ (({𝑥} ∈ 𝒫 𝐴 ∧ {𝑦} ∈ 𝒫 𝐴 ∧ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)) → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
27	6, 11, 13, 15, 17, 26	syl113anc 1330	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
28	27	ex 449	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
29	28	ralrimivva 2954	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
30		unipw 4845	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
31	30	eqcomi 2619	. . 3 ⊢ 𝐴 = ∪ 𝒫 𝐴
32	31	ishaus 20936	. 2 ⊢ (𝒫 𝐴 ∈ Haus ↔ (𝒫 𝐴 ∈ Top ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))))
33	1, 29, 32	sylanbrc 695	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  Topctop 20517  Hauscha 20922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-top 20521  df-haus 20929

This theorem is referenced by:  ssoninhaus  31617