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Theorem maxlp 23038
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 4000 . . . . 5 𝑋 βŠ† 𝑋
2 lpfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
32lpss 23033 . . . . 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 3977 . . 3 (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) β†’ 𝑃 ∈ 𝑋))
65pm4.71rd 562 . 2 (𝐽 ∈ Top β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ (𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹))))
7 simpl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ 𝐽 ∈ Top)
82islp 23031 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋 βŠ† 𝑋) β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ 𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃}))))
97, 1, 8sylancl 585 . . . 4 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((limPtβ€˜π½)β€˜π‘‹) ↔ 𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃}))))
10 snssi 4807 . . . . . 6 (𝑃 ∈ 𝑋 β†’ {𝑃} βŠ† 𝑋)
112clsdif 22944 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) = (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})))
1210, 11sylan2 592 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) = (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})))
1312eleq2d 2814 . . . 4 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜(𝑋 βˆ– {𝑃})) ↔ 𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃}))))
14 eldif 3954 . . . . . . 7 (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ (𝑃 ∈ 𝑋 ∧ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
1514baib 535 . . . . . 6 (𝑃 ∈ 𝑋 β†’ (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
1615adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ (𝑃 ∈ (𝑋 βˆ– ((intβ€˜π½)β€˜{𝑃})) ↔ Β¬ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})))
17 snssi 4807 . . . . . . . . . 10 (𝑃 ∈ ((intβ€˜π½)β€˜{𝑃}) β†’ {𝑃} βŠ† ((intβ€˜π½)β€˜{𝑃}))
1817adantl 481 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} βŠ† ((intβ€˜π½)β€˜{𝑃}))
192ntrss2 22948 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2010, 19sylan2 592 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2120adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ ((intβ€˜π½)β€˜{𝑃}) βŠ† {𝑃})
2218, 21eqssd 3995 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} = ((intβ€˜π½)β€˜{𝑃}))
232ntropn 22940 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2410, 23sylan2 592 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2524adantr 480 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ ((intβ€˜π½)β€˜{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2828 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((intβ€˜π½)β€˜{𝑃})) β†’ {𝑃} ∈ 𝐽)
27 snidg 4658 . . . . . . . . 9 (𝑃 ∈ 𝑋 β†’ 𝑃 ∈ {𝑃})
2827ad2antlr 726 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) β†’ 𝑃 ∈ {𝑃})
29 isopn3i 22973 . . . . . . . . 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 2831 . . . . . . 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 578 . 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 395   = wceq 1534   ∈ wcel 2099   βˆ– cdif 3941   βŠ† wss 3944  {csn 4624  βˆͺ cuni 4903  β€˜cfv 6542  Topctop 22782  intcnt 22908  clsccl 22909  limPtclp 23025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22783  df-cld 22910  df-ntr 22911  df-cls 22912  df-lp 23027
This theorem is referenced by:  isperf3  23044
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