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Theorem nlpineqsn 36277
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 = βˆͺ 𝐽
Assertion
Ref Expression
nlpineqsn ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ βˆ€π‘ ∈ 𝐴 βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
Distinct variable groups:   𝐴,𝑛,𝑝   𝑛,𝐽,𝑝   𝑛,𝑋,𝑝

Proof of Theorem nlpineqsn
StepHypRef Expression
1 simp1 1136 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐽 ∈ Top)
2 simp2 1137 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐴 βŠ† 𝑋)
3 ssel2 3976 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
433adant1 1130 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
51, 2, 43jca 1128 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ (𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋))
6 noel 4329 . . . . . . . . 9 Β¬ 𝑝 ∈ βˆ…
7 eleq2 2822 . . . . . . . . 9 (((limPtβ€˜π½)β€˜π΄) = βˆ… β†’ (𝑝 ∈ ((limPtβ€˜π½)β€˜π΄) ↔ 𝑝 ∈ βˆ…))
86, 7mtbiri 326 . . . . . . . 8 (((limPtβ€˜π½)β€˜π΄) = βˆ… β†’ Β¬ 𝑝 ∈ ((limPtβ€˜π½)β€˜π΄))
98adantl 482 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ Β¬ 𝑝 ∈ ((limPtβ€˜π½)β€˜π΄))
10 nlpineqsn.x . . . . . . . . 9 𝑋 = βˆͺ 𝐽
1110islp3 22641 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋) β†’ (𝑝 ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘› ∈ 𝐽 (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…)))
1211adantr 481 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ (𝑝 ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘› ∈ 𝐽 (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…)))
139, 12mtbid 323 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ Β¬ βˆ€π‘› ∈ 𝐽 (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
14 nne 2944 . . . . . . . . . 10 (Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ… ↔ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…)
1514anbi2i 623 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…))
16 annim 404 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
1715, 16bitr3i 276 . . . . . . . 8 ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
1817rexbii 3094 . . . . . . 7 (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
19 rexnal 3100 . . . . . . 7 (βˆƒπ‘› ∈ 𝐽 Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ Β¬ βˆ€π‘› ∈ 𝐽 (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
2018, 19bitri 274 . . . . . 6 (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ Β¬ βˆ€π‘› ∈ 𝐽 (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
2113, 20sylibr 233 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…))
225, 21sylan 580 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…))
23 indif2 4269 . . . . . . . . . . . 12 (𝑛 ∩ (𝐴 βˆ– {𝑝})) = ((𝑛 ∩ 𝐴) βˆ– {𝑝})
2423eqeq1i 2737 . . . . . . . . . . 11 ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ ((𝑛 ∩ 𝐴) βˆ– {𝑝}) = βˆ…)
25 ssdif0 4362 . . . . . . . . . . 11 ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ ((𝑛 ∩ 𝐴) βˆ– {𝑝}) = βˆ…)
2624, 25bitr4i 277 . . . . . . . . . 10 ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) βŠ† {𝑝})
27 elin 3963 . . . . . . . . . . 11 (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28 sssn 4828 . . . . . . . . . . . 12 ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ ((𝑛 ∩ 𝐴) = βˆ… ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29 n0i 4332 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑛 ∩ 𝐴) β†’ Β¬ (𝑛 ∩ 𝐴) = βˆ…)
30 biorf 935 . . . . . . . . . . . . 13 (Β¬ (𝑛 ∩ 𝐴) = βˆ… β†’ ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = βˆ… ∨ (𝑛 ∩ 𝐴) = {𝑝})))
3129, 30syl 17 . . . . . . . . . . . 12 (𝑝 ∈ (𝑛 ∩ 𝐴) β†’ ((𝑛 ∩ 𝐴) = {𝑝} ↔ ((𝑛 ∩ 𝐴) = βˆ… ∨ (𝑛 ∩ 𝐴) = {𝑝})))
3228, 31bitr4id 289 . . . . . . . . . . 11 (𝑝 ∈ (𝑛 ∩ 𝐴) β†’ ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3327, 32sylbir 234 . . . . . . . . . 10 ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) β†’ ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3426, 33bitrid 282 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) β†’ ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3534ancoms 459 . . . . . . . 8 ((𝑝 ∈ 𝐴 ∧ 𝑝 ∈ 𝑛) β†’ ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3635pm5.32da 579 . . . . . . 7 (𝑝 ∈ 𝐴 β†’ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
3736rexbidv 3178 . . . . . 6 (𝑝 ∈ 𝐴 β†’ (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38373ad2ant3 1135 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
3938adantr 481 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
4022, 39mpbid 231 . . 3 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
41403an1rs 1359 . 2 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) ∧ 𝑝 ∈ 𝐴) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
4241ralrimiva 3146 1 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) β†’ βˆ€π‘ ∈ 𝐴 βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6540  Topctop 22386  limPtclp 22629
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22387  df-cld 22514  df-ntr 22515  df-cls 22516  df-lp 22631
This theorem is referenced by:  nlpfvineqsn  36278  pibt2  36286
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