Theorem dishaus 23400

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 23012	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		simplrl 775	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝐴)
3	2	snssd 4821	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑥} ⊆ 𝐴)
4		vsnex 5439	. . . . . . 7 ⊢ {𝑥} ∈ V
5	4	elpw 4614	. . . . . 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 776	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝐴)
8	7	snssd 4821	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → {𝑦} ⊆ 𝐴)
9		vsnex 5439	. . . . . . 7 ⊢ {𝑦} ∈ V
10	9	elpw 4614	. . . . . 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 4673	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
13	12	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ {𝑥})
14		vsnid 4673	. . . . . 6 ⊢ 𝑦 ∈ {𝑦}
15	14	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ {𝑦})
16		disjsn2 4724	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 → ({𝑥} ∩ {𝑦}) = ∅)
17	16	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ({𝑥} ∩ {𝑦}) = ∅)
18		eleq2 2818	. . . . . . 7 ⊢ (𝑢 = {𝑥} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑥}))
19		ineq1 4216	. . . . . . . 8 ⊢ (𝑢 = {𝑥} → (𝑢 ∩ 𝑣) = ({𝑥} ∩ 𝑣))
20	19	eqeq1d 2731	. . . . . . 7 ⊢ (𝑢 = {𝑥} → ((𝑢 ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ 𝑣) = ∅))
21	18, 20	3anbi13d 1435	. . . . . 6 ⊢ (𝑢 = {𝑥} → ((𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅)))
22		eleq2 2818	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ {𝑦}))
23		ineq2 4217	. . . . . . . 8 ⊢ (𝑣 = {𝑦} → ({𝑥} ∩ 𝑣) = ({𝑥} ∩ {𝑦}))
24	23	eqeq1d 2731	. . . . . . 7 ⊢ (𝑣 = {𝑦} → (({𝑥} ∩ 𝑣) = ∅ ↔ ({𝑥} ∩ {𝑦}) = ∅))
25	22, 24	3anbi23d 1436	. . . . . 6 ⊢ (𝑣 = {𝑦} → ((𝑥 ∈ {𝑥} ∧ 𝑦 ∈ 𝑣 ∧ ({𝑥} ∩ 𝑣) = ∅) ↔ (𝑥 ∈ {𝑥} ∧ 𝑦 ∈ {𝑦} ∧ ({𝑥} ∩ {𝑦}) = ∅)))
26	21, 25	rspc2ev 3633	. . . . 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 1379	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ≠ 𝑦) → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
28	27	ex 411	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
29	28	ralrimivva 3195	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → ∃𝑢 ∈ 𝒫 𝐴∃𝑣 ∈ 𝒫 𝐴(𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅)))
30		unipw 5460	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
31	30	eqcomi 2738	. . 3 ⊢ 𝐴 = ∪ 𝒫 𝐴
32	31	ishaus 23340	. 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 581	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Haus)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100   ≠ wne 2933  ∀wral 3054  ∃wrex 3063   ∩ cin 3958   ⊆ wss 3959  ∅c0 4335  𝒫 cpw 4610  {csn 4636  ∪ cuni 4918  Topctop 22909  Hauscha 23326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5307  ax-pow 5373  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-pw 4612  df-sn 4637  df-pr 4639  df-uni 4919  df-top 22910  df-haus 23333

This theorem is referenced by:  ssoninhaus  36228