Theorem nlpineqsn 35579

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 1135	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐽 ∈ Top)
2		simp2 1136	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝐴 ⊆ 𝑋)
3		ssel2 3916	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
4	3	3adant1 1129	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝑋)
5	1, 2, 4	3jca 1127	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋))
6		noel 4264	. . . . . . . . 9 ⊢ ¬ 𝑝 ∈ ∅
7		eleq2 2827	. . . . . . . . 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 482	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
10		nlpineqsn.x	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 22297	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑋) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
12	11	adantr 481	. . . . . . 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 2947	. . . . . . . . . 10 ⊢ (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
15	14	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
16		annim 404	. . . . . . . . 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 3181	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19		rexnal 3169	. . . . . . 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 580	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23		indif2 4204	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛 ∩ 𝐴) ∖ {𝑝})
24	23	eqeq1i 2743	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
25		ssdif0 4297	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
26	24, 25	bitr4i 277	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) ⊆ {𝑝})
27		elin 3903	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28		sssn 4759	. . . . . . . . . . . 12 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29		n0i 4267	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) → ¬ (𝑛 ∩ 𝐴) = ∅)
30		biorf 934	. . . . . . . . . . . . 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 234	. . . . . . . . . 10 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
34	26, 33	syl5bb 283	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) = {𝑝}))
35	34	ancoms 459	. . . . . . . 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 3226	. . . . . 6 ⊢ (𝑝 ∈ 𝐴 → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38	37	3ad2ant3 1134	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) → (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
39	38	adantr 481	. . . 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 1358	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝 ∈ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
42	41	ralrimiva 3103	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 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  limPtclp 22285

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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  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-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172  df-lp 22287

This theorem is referenced by:  nlpfvineqsn  35580  pibt2  35588