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Theorem maxlp 22973
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 = 𝐽
Assertion
Ref Expression
maxlp (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))

Proof of Theorem maxlp
StepHypRef Expression
1 ssid 3996 . . . . 5 𝑋𝑋
2 lpfval.1 . . . . . 6 𝑋 = 𝐽
32lpss 22968 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
41, 3mpan2 688 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
54sseld 3973 . . 3 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃𝑋))
65pm4.71rd 562 . 2 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7 simpl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → 𝐽 ∈ Top)
82islp 22966 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
97, 1, 8sylancl 585 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10 snssi 4803 . . . . . 6 (𝑃𝑋 → {𝑃} ⊆ 𝑋)
112clsdif 22879 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1210, 11sylan2 592 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1312eleq2d 2811 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14 eldif 3950 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1514baib 535 . . . . . 6 (𝑃𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1615adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17 snssi 4803 . . . . . . . . . 10 (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
1817adantl 481 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
192ntrss2 22883 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2010, 19sylan2 592 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2120adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2218, 21eqssd 3991 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
232ntropn 22875 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2410, 23sylan2 592 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2524adantr 480 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2825 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27 snidg 4654 . . . . . . . . 9 (𝑃𝑋𝑃 ∈ {𝑃})
2827ad2antlr 724 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29 isopn3i 22908 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3029adantlr 712 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3128, 30eleqtrrd 2828 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
3226, 31impbida 798 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
3332notbid 318 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
3416, 33bitrd 279 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
359, 13, 343bitrd 305 . . 3 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
3635pm5.32da 578 . 2 (𝐽 ∈ Top → ((𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
376, 36bitrd 279 1 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cdif 3937  wss 3940  {csn 4620   cuni 4899  cfv 6533  Topctop 22717  intcnt 22843  clsccl 22844  limPtclp 22960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 22718  df-cld 22845  df-ntr 22846  df-cls 22847  df-lp 22962
This theorem is referenced by:  isperf3  22979
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