Theorem nlpineqsn 37409

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 1137	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐽 ∈ Top)
2		simp2 1138	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐴 ⊆ 𝑋)
3		ssel2 3978	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
4	3	3adant1 1131	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
5	1, 2, 4	3jca 1129	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋))
6		noel 4338	. . . . . . . . 9 ⊢ ¬ 𝑝 ∈ ∅
7		eleq2 2830	. . . . . . . . 9 ⊢ (((limPt‘𝐽)‘𝐴) = ∅ → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ 𝑝 ∈ ∅))
8	6, 7	mtbiri 327	. . . . . . . 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 23154	. . . . . . . 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 324	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
14		nne 2944	. . . . . . . . . 10 ⊢ (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
15	14	anbi2i 623	. . . . . . . . 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 277	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
18	17	rexbii 3094	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19		rexnal 3100	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
20	18, 19	bitri 275	. . . . . 6 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
21	13, 20	sylibr 234	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
22	5, 21	sylan 580	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23		indif2 4281	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛 ∩ 𝐴) ∖ {𝑝})
24	23	eqeq1i 2742	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
25		ssdif0 4366	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
26	24, 25	bitr4i 278	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) ⊆ {𝑝})
27		elin 3967	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28		sssn 4826	. . . . . . . . . . . 12 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29		n0i 4340	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ¬ (𝑛 ∩ 𝐴) = ∅)
30		biorf 937	. . . . . . . . . . . . 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 290	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
33	27, 32	sylbir 235	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
34	26, 33	bitrid 283	. . . . . . . . 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 579	. . . . . . 7 ⊢ (𝑝 ∈ 𝐴 → ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
37	36	rexbidv 3179	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38	37	3ad2ant3 1136	. . . . 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 232	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
41	40	3an1rs 1360	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝 ∈ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
42	41	ralrimiva 3146	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  limPtclp 23142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029  df-lp 23144

This theorem is referenced by:  nlpfvineqsn  37410  pibt2  37418