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Theorem nlpineqsn 36593
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 1135 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐽 ∈ Top)
2 simp2 1136 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐴 βŠ† 𝑋)
3 ssel2 3978 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
433adant1 1129 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
51, 2, 43jca 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ (𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋))
6 noel 4331 . . . . . . . . 9 Β¬ 𝑝 ∈ βˆ…
7 eleq2 2821 . . . . . . . . 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 481 . . . . . . 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 22871 . . . . . . . 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 480 . . . . . . 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 2943 . . . . . . . . . 10 (Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ… ↔ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…)
1514anbi2i 622 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…))
16 annim 403 . . . . . . . . 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 3093 . . . . . . 7 (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
19 rexnal 3099 . . . . . . 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 579 . . . 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 4271 . . . . . . . . . . . 12 (𝑛 ∩ (𝐴 βˆ– {𝑝})) = ((𝑛 ∩ 𝐴) βˆ– {𝑝})
2423eqeq1i 2736 . . . . . . . . . . 11 ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ ((𝑛 ∩ 𝐴) βˆ– {𝑝}) = βˆ…)
25 ssdif0 4364 . . . . . . . . . . 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 3965 . . . . . . . . . . 11 (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28 sssn 4830 . . . . . . . . . . . 12 ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ ((𝑛 ∩ 𝐴) = βˆ… ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29 n0i 4334 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑛 ∩ 𝐴) β†’ Β¬ (𝑛 ∩ 𝐴) = βˆ…)
30 biorf 934 . . . . . . . . . . . . 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 458 . . . . . . . 8 ((𝑝 ∈ 𝐴 ∧ 𝑝 ∈ 𝑛) β†’ ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3635pm5.32da 578 . . . . . . 7 (𝑝 ∈ 𝐴 β†’ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
3736rexbidv 3177 . . . . . 6 (𝑝 ∈ 𝐴 β†’ (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38373ad2ant3 1134 . . . . 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 480 . . . 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 1358 . 2 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) ∧ 𝑝 ∈ 𝐴) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
4241ralrimiva 3145 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 395   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22616  limPtclp 22859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22617  df-cld 22744  df-ntr 22745  df-cls 22746  df-lp 22861
This theorem is referenced by:  nlpfvineqsn  36594  pibt2  36602
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