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Theorem maxlp 22651
Description: A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
maxlp (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ (𝑃 ∈ 𝑋 ∧ Β¬ {𝑃} ∈ 𝐽)))

Proof of Theorem maxlp
StepHypRef Expression
1 ssid 4005 . . . . 5 𝑋 βŠ† 𝑋
2 lpfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32lpss 22646 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋 βŠ† 𝑋) β†’ ((limPtβ€˜π½)β€˜π‘‹) βŠ† 𝑋)
41, 3mpan2 690 . . . 4 (𝐽 ∈ Top β†’ ((limPtβ€˜π½)β€˜π‘‹) βŠ† 𝑋)
54sseld 3982 . . 3 (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) β†’ 𝑃 ∈ 𝑋))
65pm4.71rd 564 . 2 (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ (𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹))))
7 simpl 484 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ 𝐽 ∈ Top)
82islp 22644 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋 βŠ† 𝑋) β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ 𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃}))))
97, 1, 8sylancl 587 . . . 4 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ 𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃}))))
10 snssi 4812 . . . . . 6 (𝑃 ∈ 𝑋 β†’ {𝑃} βŠ† 𝑋)
112clsdif 22557 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) = (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})))
1210, 11sylan2 594 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) = (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})))
1312eleq2d 2820 . . . 4 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) ↔ 𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃}))))
14 eldif 3959 . . . . . . 7 (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ (𝑃 ∈ 𝑋 ∧ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
1514baib 537 . . . . . 6 (𝑃 ∈ 𝑋 β†’ (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
1615adantl 483 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
17 snssi 4812 . . . . . . . . . 10 (𝑃 ∈ ((intβ€˜π½)β€˜{𝑃}) β†’ {𝑃} βŠ† ((intβ€˜π½)β€˜{𝑃}))
1817adantl 483 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} βŠ† ((intβ€˜π½)β€˜{𝑃}))
192ntrss2 22561 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2010, 19sylan2 594 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2120adantr 482 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2218, 21eqssd 4000 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} = ((intβ€˜π½)β€˜{𝑃}))
232ntropn 22553 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2410, 23sylan2 594 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2524adantr 482 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2834 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} ∈ 𝐽)
27 snidg 4663 . . . . . . . . 9 (𝑃 ∈ 𝑋 β†’ 𝑃 ∈ {𝑃})
2827ad2antlr 726 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) β†’ 𝑃 ∈ {𝑃})
29 isopn3i 22586 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) β†’ ((intβ€˜π½)β€˜{𝑃}) = {𝑃})
3029adantlr 714 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) β†’ ((intβ€˜π½)β€˜{𝑃}) = {𝑃})
3128, 30eleqtrrd 2837 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) β†’ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃}))
3226, 31impbida 800 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((intβ€˜π½)β€˜{𝑃}) ↔ {𝑃} ∈ 𝐽))
3332notbid 318 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃}) ↔ Β¬ {𝑃} ∈ 𝐽))
3416, 33bitrd 279 . . . 4 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ Β¬ {𝑃} ∈ 𝐽))
359, 13, 343bitrd 305 . . 3 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ Β¬ {𝑃} ∈ 𝐽))
3635pm5.32da 580 . 2 (𝐽 ∈ Top β†’ ((𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹)) ↔ (𝑃 ∈ 𝑋 ∧ Β¬ {𝑃} ∈ 𝐽)))
376, 36bitrd 279 1 (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ (𝑃 ∈ 𝑋 ∧ Β¬ {𝑃} ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  intcnt 22521  clsccl 22522  limPtclp 22638
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 22396  df-cld 22523  df-ntr 22524  df-cls 22525  df-lp 22640
This theorem is referenced by:  isperf3  22657
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