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Theorem nlpineqsn 34976
 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 ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝𝐴𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
Distinct variable groups:   𝐴,𝑛,𝑝   𝑛,𝐽,𝑝   𝑛,𝑋,𝑝

Proof of Theorem nlpineqsn
StepHypRef Expression
1 simp1 1133 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝐽 ∈ Top)
2 simp2 1134 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝐴𝑋)
3 ssel2 3912 . . . . . . 7 ((𝐴𝑋𝑝𝐴) → 𝑝𝑋)
433adant1 1127 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝑝𝑋)
51, 2, 43jca 1125 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → (𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋))
6 noel 4250 . . . . . . . . 9 ¬ 𝑝 ∈ ∅
7 eleq2 2878 . . . . . . . . 9 (((limPt‘𝐽)‘𝐴) = ∅ → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ 𝑝 ∈ ∅))
86, 7mtbiri 330 . . . . . . . 8 (((limPt‘𝐽)‘𝐴) = ∅ → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
98adantl 485 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
10 nlpineqsn.x . . . . . . . . 9 𝑋 = 𝐽
1110islp3 21792 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
1211adantr 484 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
139, 12mtbid 327 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
14 nne 2991 . . . . . . . . . 10 (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
1514anbi2i 625 . . . . . . . . 9 ((𝑝𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
16 annim 407 . . . . . . . . 9 ((𝑝𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
1715, 16bitr3i 280 . . . . . . . 8 ((𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
1817rexbii 3211 . . . . . . 7 (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19 rexnal 3201 . . . . . . 7 (∃𝑛𝐽 ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
2018, 19bitri 278 . . . . . 6 (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
2113, 20sylibr 237 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
225, 21sylan 583 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23 indif2 4200 . . . . . . . . . . . 12 (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛𝐴) ∖ {𝑝})
2423eqeq1i 2803 . . . . . . . . . . 11 ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛𝐴) ∖ {𝑝}) = ∅)
25 ssdif0 4280 . . . . . . . . . . 11 ((𝑛𝐴) ⊆ {𝑝} ↔ ((𝑛𝐴) ∖ {𝑝}) = ∅)
2624, 25bitr4i 281 . . . . . . . . . 10 ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) ⊆ {𝑝})
27 elin 3899 . . . . . . . . . . 11 (𝑝 ∈ (𝑛𝐴) ↔ (𝑝𝑛𝑝𝐴))
28 sssn 4722 . . . . . . . . . . . 12 ((𝑛𝐴) ⊆ {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝}))
29 n0i 4252 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑛𝐴) → ¬ (𝑛𝐴) = ∅)
30 biorf 934 . . . . . . . . . . . . 13 (¬ (𝑛𝐴) = ∅ → ((𝑛𝐴) = {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝})))
3129, 30syl 17 . . . . . . . . . . . 12 (𝑝 ∈ (𝑛𝐴) → ((𝑛𝐴) = {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝})))
3228, 31bitr4id 293 . . . . . . . . . . 11 (𝑝 ∈ (𝑛𝐴) → ((𝑛𝐴) ⊆ {𝑝} ↔ (𝑛𝐴) = {𝑝}))
3327, 32sylbir 238 . . . . . . . . . 10 ((𝑝𝑛𝑝𝐴) → ((𝑛𝐴) ⊆ {𝑝} ↔ (𝑛𝐴) = {𝑝}))
3426, 33syl5bb 286 . . . . . . . . 9 ((𝑝𝑛𝑝𝐴) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) = {𝑝}))
3534ancoms 462 . . . . . . . 8 ((𝑝𝐴𝑝𝑛) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) = {𝑝}))
3635pm5.32da 582 . . . . . . 7 (𝑝𝐴 → ((𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
3736rexbidv 3257 . . . . . 6 (𝑝𝐴 → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
38373ad2ant3 1132 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
3938adantr 484 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
4022, 39mpbid 235 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
41403an1rs 1356 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝𝐴) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
4241ralrimiva 3149 1 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝𝐴𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4246  {csn 4528  ∪ cuni 4804  ‘cfv 6332  Topctop 21539  limPtclp 21780 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-top 21540  df-cld 21665  df-ntr 21666  df-cls 21667  df-lp 21782 This theorem is referenced by:  nlpfvineqsn  34977  pibt2  34985
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