Theorem nlpineqsn 37720

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 3917	. . . . . . 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 4279	. . . . . . . . 9 ⊢ ¬ 𝑝 ∈ ∅
7		eleq2 2826	. . . . . . . . 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 23089	. . . . . . . 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 2937	. . . . . . . . . 10 ⊢ (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
15	14	anbi2i 624	. . . . . . . . 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 3085	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛 ∈ 𝐽 ¬ (𝑝 ∈ 𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19		rexnal 3090	. . . . . . 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 581	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23		indif2 4222	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛 ∩ 𝐴) ∖ {𝑝})
24	23	eqeq1i 2742	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
25		ssdif0 4307	. . . . . . . . . . 11 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) ∖ {𝑝}) = ∅)
26	24, 25	bitr4i 278	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛 ∩ 𝐴) ⊆ {𝑝})
27		elin 3906	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28		sssn 4770	. . . . . . . . . . . 12 ⊢ ((𝑛 ∩ 𝐴) ⊆ {𝑝} ↔ ((𝑛 ∩ 𝐴) = ∅ ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29		n0i 4281	. . . . . . . . . . . . 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 3162	. . . . . 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 1361	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝 ∈ 𝐴) → ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
42	41	ralrimiva 3130	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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ‘cfv 6490  Topctop 22836  limPtclp 23077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-top 22837  df-cld 22962  df-ntr 22963  df-cls 22964  df-lp 23079

This theorem is referenced by:  nlpfvineqsn  37721  pibt2  37729