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Theorem nlpineqsn 35925
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 1137 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐽 ∈ Top)
2 simp2 1138 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝐴 βŠ† 𝑋)
3 ssel2 3940 . . . . . . 7 ((𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
433adant1 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝑋)
51, 2, 43jca 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝐴) β†’ (𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝑝 ∈ 𝑋))
6 noel 4291 . . . . . . . . 9 Β¬ 𝑝 ∈ βˆ…
7 eleq2 2823 . . . . . . . . 9 (((limPtβ€˜π½)β€˜π΄) = βˆ… β†’ (𝑝 ∈ ((limPtβ€˜π½)β€˜π΄) ↔ 𝑝 ∈ βˆ…))
86, 7mtbiri 327 . . . . . . . 8 (((limPtβ€˜π½)β€˜π΄) = βˆ… β†’ Β¬ 𝑝 ∈ ((limPtβ€˜π½)β€˜π΄))
98adantl 483 . . . . . . 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 22513 . . . . . . . 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 482 . . . . . . 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 324 . . . . . 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 624 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…))
16 annim 405 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ Β¬ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…) ↔ Β¬ (𝑝 ∈ 𝑛 β†’ (𝑛 ∩ (𝐴 βˆ– {𝑝})) β‰  βˆ…))
1715, 16bitr3i 277 . . . . . . . 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 275 . . . . . 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 581 . . . 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 4231 . . . . . . . . . . . 12 (𝑛 ∩ (𝐴 βˆ– {𝑝})) = ((𝑛 ∩ 𝐴) βˆ– {𝑝})
2423eqeq1i 2738 . . . . . . . . . . 11 ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ ((𝑛 ∩ 𝐴) βˆ– {𝑝}) = βˆ…)
25 ssdif0 4324 . . . . . . . . . . 11 ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ ((𝑛 ∩ 𝐴) βˆ– {𝑝}) = βˆ…)
2624, 25bitr4i 278 . . . . . . . . . 10 ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) βŠ† {𝑝})
27 elin 3927 . . . . . . . . . . 11 (𝑝 ∈ (𝑛 ∩ 𝐴) ↔ (𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴))
28 sssn 4787 . . . . . . . . . . . 12 ((𝑛 ∩ 𝐴) βŠ† {𝑝} ↔ ((𝑛 ∩ 𝐴) = βˆ… ∨ (𝑛 ∩ 𝐴) = {𝑝}))
29 n0i 4294 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑛 ∩ 𝐴) β†’ Β¬ (𝑛 ∩ 𝐴) = βˆ…)
30 biorf 936 . . . . . . . . . . . . 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 290 . . . . . . . . . . 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 283 . . . . . . . . 9 ((𝑝 ∈ 𝑛 ∧ 𝑝 ∈ 𝐴) β†’ ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3534ancoms 460 . . . . . . . 8 ((𝑝 ∈ 𝐴 ∧ 𝑝 ∈ 𝑛) β†’ ((𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ… ↔ (𝑛 ∩ 𝐴) = {𝑝}))
3635pm5.32da 580 . . . . . . 7 (𝑝 ∈ 𝐴 β†’ ((𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
3736rexbidv 3172 . . . . . 6 (𝑝 ∈ 𝐴 β†’ (βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ (𝐴 βˆ– {𝑝})) = βˆ…) ↔ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})))
38373ad2ant3 1136 . . . . 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 482 . . . 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 1360 . 2 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ ((limPtβ€˜π½)β€˜π΄) = βˆ…) ∧ 𝑝 ∈ 𝐴) β†’ βˆƒπ‘› ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝}))
4241ralrimiva 3140 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3908   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  {csn 4587  βˆͺ cuni 4866  β€˜cfv 6497  Topctop 22258  limPtclp 22501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-top 22259  df-cld 22386  df-ntr 22387  df-cls 22388  df-lp 22503
This theorem is referenced by:  nlpfvineqsn  35926  pibt2  35934
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