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Theorem maxlp 21731
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 3968 . . . . 5 𝑋𝑋
2 lpfval.1 . . . . . 6 𝑋 = 𝐽
32lpss 21726 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
41, 3mpan2 689 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
54sseld 3945 . . 3 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃𝑋))
65pm4.71rd 565 . 2 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7 simpl 485 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → 𝐽 ∈ Top)
82islp 21724 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
97, 1, 8sylancl 588 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10 snssi 4717 . . . . . 6 (𝑃𝑋 → {𝑃} ⊆ 𝑋)
112clsdif 21637 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1210, 11sylan2 594 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1312eleq2d 2896 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14 eldif 3923 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1514baib 538 . . . . . 6 (𝑃𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1615adantl 484 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17 snssi 4717 . . . . . . . . . 10 (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
1817adantl 484 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
192ntrss2 21641 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2010, 19sylan2 594 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2120adantr 483 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2218, 21eqssd 3963 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
232ntropn 21633 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2410, 23sylan2 594 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2524adantr 483 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2911 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27 snidg 4575 . . . . . . . . 9 (𝑃𝑋𝑃 ∈ {𝑃})
2827ad2antlr 725 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29 isopn3i 21666 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3029adantlr 713 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3128, 30eleqtrrd 2914 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
3226, 31impbida 799 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
3332notbid 320 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
3416, 33bitrd 281 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
359, 13, 343bitrd 307 . . 3 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
3635pm5.32da 581 . 2 (𝐽 ∈ Top → ((𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
376, 36bitrd 281 1 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  cdif 3910  wss 3913  {csn 4543   cuni 4814  cfv 6331  Topctop 21477  intcnt 21601  clsccl 21602  limPtclp 21718
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-top 21478  df-cld 21603  df-ntr 21604  df-cls 21605  df-lp 21720
This theorem is referenced by:  isperf3  21737
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