Theorem nlpineqsn 38087

Description: For every point 𝑝 of a subset 𝐴 of 𝑋 with no limit points, there exists an open set 𝑛 that intersects 𝐴 only at 𝑝. (Contributed by ML, 23-Mar-2021.)

Hypothesis

Ref	Expression
nlpineqsn.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
nlpineqsn	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))

Distinct variable groups:   𝐴,𝑛,𝑝   𝑛,𝐽,𝑝   𝑛,𝑋,𝑝

Proof of Theorem nlpineqsn

Step	Hyp	Ref	Expression
1		simp1 1154	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐽 ∈ Top)
2		simp2 1155	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐴 ⊆ 𝑋)
3		ssel2 3935	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
4	3	3adant1 1148	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
5	1, 2, 4	3jca 1146	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋))
6		noel 4294	. . . . . . . . 9 ⊢ ¬ 𝑝 ∈ ∅
7		eleq2 2855	. . . . . . . . 9 ⊢ (((limPt‘𝐽)‘𝐴) = ∅ → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ 𝑝 ∈ ∅))
8	6, 7	mtbiri 330	. . . . . . . 8 ⊢ (((limPt‘𝐽)‘𝐴) = ∅ → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
9	8	adantl 487	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
10		nlpineqsn.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 23333	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
12	11	adantr 486	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
13	9, 12	mtbid 327	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
14		nne 2965	. . . . . . . . . 10 ⊢ (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
15	14	anbi2i 635	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
16		annim 409	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
17	15, 16	bitr3i 280	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
18	17	rexbii 3115	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19		rexnal 3120	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
20	18, 19	bitri 278	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
21	13, 20	sylibr 237	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
22	5, 21	sylan 592	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23		indif2 4237	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛 ∩ 𝐴) ∖ {𝑝})
24	23	eqeq1i 2771	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
25		ssdif0 4324	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
26	24, 25	bitr4i 281	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) ⊆ {𝑝})
27		elin 3924	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28		sssn 4797	. . . . . . . . . . . 12 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29		n0i 4296	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ¬ (𝑛 ∩ 𝐴) = ∅)
30		biorf 950	. . . . . . . . . . . . 13 ⊢ (¬ (𝑛 ∩ 𝐴) = ∅ → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝})))
31	29, 30	syl 18	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝})))
32	28, 31	bitr4id 293	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
33	27, 32	sylbir 238	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
34	26, 33	bitrid 286	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) = {𝑝}))
35	34	ancoms 464	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝 ∈ 𝑛) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) = {𝑝}))
36	35	pm5.32da 590	. . . . . . 7 ⊢ (𝑝 ∈ 𝐴 → ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
37	36	rexbidv 3192	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38	37	3ad2ant3 1153	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
39	38	adantr 486	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
40	22, 39	mpbid 235	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
41	40	3an1rs 1378	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝 ∈ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
42	41	ralrimiva 3160	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4594  ∪ cuni 4877  ‘cfv 6543  Topctop 23080  limPtclp 23321

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 23081  df-cld 23206  df-ntr 23207  df-cls 23208  df-lp 23323

This theorem is used by:  nlpfvineqsn  38088  pibt2  38096