Theorem dishaus 22533

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 22145	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		simplrl 774	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝐴)
3	2	snssd 4742	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑥} ⊆ 𝐴)
4		snex 5354	. . . . . . 7 ⊢ {𝑥} ∈ V
5	4	elpw 4537	. . . . . 6 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
6	3, 5	sylibr 233	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑥} ∈ 𝒫 𝐴)
7		simplrr 775	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝐴)
8	7	snssd 4742	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑦} ⊆ 𝐴)
9		snex 5354	. . . . . . 7 ⊢ {𝑦} ∈ V
10	9	elpw 4537	. . . . . 6 ⊢ ({𝑦} ∈ 𝒫 𝐴 ↔ {𝑦} ⊆ 𝐴)
11	8, 10	sylibr 233	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑦} ∈ 𝒫 𝐴)
12		vsnid 4598	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
13	12	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ {𝑥})
14		vsnid 4598	. . . . . 6 ⊢ 𝑦 ∈ {𝑦}
15	14	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ {𝑦})
16		disjsn2 4648	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
17	16	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18		eleq2 2827	. . . . . . 7 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
19		ineq1 4139	. . . . . . . 8 ⊢ (𝑢 = {𝑥} → (𝑢 ∩ 𝑣) = ({𝑥} ∩ 𝑣))
20	19	eqeq1d 2740	. . . . . . 7 ⊢ (𝑢 = {𝑥} → ((𝑢 ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
21	18, 20	3anbi13d 1437	. . . . . 6 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22		eleq2 2827	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ {𝑦}))
23		ineq2 4140	. . . . . . . 8 ⊢ (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
24	23	eqeq1d 2740	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
25	22, 24	3anbi23d 1438	. . . . . 6 ⊢ (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
26	21, 25	rspc2ev 3572	. . . . 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 1381	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
28	27	ex 413	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
29	28	ralrimivva 3123	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
30		unipw 5366	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
31	30	eqcomi 2747	. . 3 ⊢ 𝐴 = ∪ 𝒫 𝐴
32	31	ishaus 22473	. 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 583	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839  Topctop 22042  Hauscha 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840  df-top 22043  df-haus 22466

This theorem is referenced by:  ssoninhaus  34637