Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nlpineqsn Structured version   Visualization version   GIF version

Theorem nlpineqsn 37391
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 1135 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝐽 ∈ Top)
2 simp2 1136 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝐴𝑋)
3 ssel2 3990 . . . . . . 7 ((𝐴𝑋𝑝𝐴) → 𝑝𝑋)
433adant1 1129 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → 𝑝𝑋)
51, 2, 43jca 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → (𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋))
6 noel 4344 . . . . . . . . 9 ¬ 𝑝 ∈ ∅
7 eleq2 2828 . . . . . . . . 9 (((limPt‘𝐽)‘𝐴) = ∅ → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ 𝑝 ∈ ∅))
86, 7mtbiri 327 . . . . . . . 8 (((limPt‘𝐽)‘𝐴) = ∅ → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
98adantl 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ 𝑝 ∈ ((limPt‘𝐽)‘𝐴))
10 nlpineqsn.x . . . . . . . . 9 𝑋 = 𝐽
1110islp3 23170 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
1211adantr 480 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (𝑝 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅)))
139, 12mtbid 324 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
14 nne 2942 . . . . . . . . . 10 (¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅ ↔ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅)
1514anbi2i 623 . . . . . . . . 9 ((𝑝𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
16 annim 403 . . . . . . . . 9 ((𝑝𝑛 ∧ ¬ (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
1715, 16bitr3i 277 . . . . . . . 8 ((𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
1817rexbii 3092 . . . . . . 7 (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
19 rexnal 3098 . . . . . . 7 (∃𝑛𝐽 ¬ (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅) ↔ ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
2018, 19bitri 275 . . . . . 6 (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ¬ ∀𝑛𝐽 (𝑝𝑛 → (𝑛 ∩ (𝐴 ∖ {𝑝})) ≠ ∅))
2113, 20sylibr 234 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝑋) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
225, 21sylan 580 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅))
23 indif2 4287 . . . . . . . . . . . 12 (𝑛 ∩ (𝐴 ∖ {𝑝})) = ((𝑛𝐴) ∖ {𝑝})
2423eqeq1i 2740 . . . . . . . . . . 11 ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ ((𝑛𝐴) ∖ {𝑝}) = ∅)
25 ssdif0 4372 . . . . . . . . . . 11 ((𝑛𝐴) ⊆ {𝑝} ↔ ((𝑛𝐴) ∖ {𝑝}) = ∅)
2624, 25bitr4i 278 . . . . . . . . . 10 ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) ⊆ {𝑝})
27 elin 3979 . . . . . . . . . . 11 (𝑝 ∈ (𝑛𝐴) ↔ (𝑝𝑛𝑝𝐴))
28 sssn 4831 . . . . . . . . . . . 12 ((𝑛𝐴) ⊆ {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝}))
29 n0i 4346 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑛𝐴) → ¬ (𝑛𝐴) = ∅)
30 biorf 936 . . . . . . . . . . . . 13 (¬ (𝑛𝐴) = ∅ → ((𝑛𝐴) = {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝})))
3129, 30syl 17 . . . . . . . . . . . 12 (𝑝 ∈ (𝑛𝐴) → ((𝑛𝐴) = {𝑝} ↔ ((𝑛𝐴) = ∅ ∨ (𝑛𝐴) = {𝑝})))
3228, 31bitr4id 290 . . . . . . . . . . 11 (𝑝 ∈ (𝑛𝐴) → ((𝑛𝐴) ⊆ {𝑝} ↔ (𝑛𝐴) = {𝑝}))
3327, 32sylbir 235 . . . . . . . . . 10 ((𝑝𝑛𝑝𝐴) → ((𝑛𝐴) ⊆ {𝑝} ↔ (𝑛𝐴) = {𝑝}))
3426, 33bitrid 283 . . . . . . . . 9 ((𝑝𝑛𝑝𝐴) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) = {𝑝}))
3534ancoms 458 . . . . . . . 8 ((𝑝𝐴𝑝𝑛) → ((𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅ ↔ (𝑛𝐴) = {𝑝}))
3635pm5.32da 579 . . . . . . 7 (𝑝𝐴 → ((𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
3736rexbidv 3177 . . . . . 6 (𝑝𝐴 → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
38373ad2ant3 1134 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
3938adantr 480 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → (∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛 ∩ (𝐴 ∖ {𝑝})) = ∅) ↔ ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝})))
4022, 39mpbid 232 . . 3 (((𝐽 ∈ Top ∧ 𝐴𝑋𝑝𝐴) ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
41403an1rs 1358 . 2 (((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) ∧ 𝑝𝐴) → ∃𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
4241ralrimiva 3144 1 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝𝐴𝑛𝐽 (𝑝𝑛 ∧ (𝑛𝐴) = {𝑝}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cdif 3960  cin 3962  wss 3963  c0 4339  {csn 4631   cuni 4912  cfv 6563  Topctop 22915  limPtclp 23158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045  df-lp 23160
This theorem is referenced by:  nlpfvineqsn  37392  pibt2  37400
  Copyright terms: Public domain W3C validator