Theorem nlpineqsn 35506

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 1134	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐽 ∈ Top)
2		simp2 1135	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐴 ⊆ 𝑋)
3		ssel2 3912	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
4	3	3adant1 1128	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
5	1, 2, 4	3jca 1126	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋))
6		noel 4261	. . . . . . . . 9 ⊢ ¬ 𝑝 ∈ ∅
7		eleq2 2827	. . . . . . . . 9 ⊢ (((limPt‘𝐽)‘𝐴) = ∅ → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ 𝑝 ∈ ∅))
8	6, 7	mtbiri 326	. . . . . . . 8 ⊢ (((limPt‘𝐽)‘𝐴) = ∅ → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
9	8	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
10		nlpineqsn.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 22205	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
12	11	adantr 480	. . . . . . 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 323	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
14		nne 2946	. . . . . . . . . 10 ⊢ (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
15	14	anbi2i 622	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
16		annim 403	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
17	15, 16	bitr3i 276	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
18	17	rexbii 3177	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19		rexnal 3165	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
20	18, 19	bitri 274	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
21	13, 20	sylibr 233	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
22	5, 21	sylan 579	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23		indif2 4201	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛 ∩ 𝐴) ∖ {𝑝})
24	23	eqeq1i 2743	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
25		ssdif0 4294	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
26	24, 25	bitr4i 277	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) ⊆ {𝑝})
27		elin 3899	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28		sssn 4756	. . . . . . . . . . . 12 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29		n0i 4264	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ¬ (𝑛 ∩ 𝐴) = ∅)
30		biorf 933	. . . . . . . . . . . . 13 ⊢ (¬ (𝑛 ∩ 𝐴) = ∅ → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝})))
31	29, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝})))
32	28, 31	bitr4id 289	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
33	27, 32	sylbir 234	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
34	26, 33	syl5bb 282	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) = {𝑝}))
35	34	ancoms 458	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝 ∈ 𝑛) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) = {𝑝}))
36	35	pm5.32da 578	. . . . . . 7 ⊢ (𝑝 ∈ 𝐴 → ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
37	36	rexbidv 3225	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38	37	3ad2ant3 1133	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
39	38	adantr 480	. . . 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 231	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
41	40	3an1rs 1357	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝 ∈ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
42	41	ralrimiva 3107	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  limPtclp 22193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-top 21951  df-cld 22078  df-ntr 22079  df-cls 22080  df-lp 22195

This theorem is referenced by:  nlpfvineqsn  35507  pibt2  35515