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Theorem maxlp 21153
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 3765 . . . . 5 𝑋𝑋
2 lpfval.1 . . . . . 6 𝑋 = 𝐽
32lpss 21148 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
41, 3mpan2 709 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
54sseld 3743 . . 3 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃𝑋))
65pm4.71rd 670 . 2 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7 simpl 474 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → 𝐽 ∈ Top)
82islp 21146 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
97, 1, 8sylancl 697 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10 snssi 4484 . . . . . 6 (𝑃𝑋 → {𝑃} ⊆ 𝑋)
112clsdif 21059 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1210, 11sylan2 492 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1312eleq2d 2825 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14 eldif 3725 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1514baib 982 . . . . . 6 (𝑃𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1615adantl 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17 snssi 4484 . . . . . . . . . 10 (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
1817adantl 473 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
192ntrss2 21063 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2010, 19sylan2 492 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2120adantr 472 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2218, 21eqssd 3761 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
232ntropn 21055 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2410, 23sylan2 492 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2524adantr 472 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2839 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27 snidg 4351 . . . . . . . . 9 (𝑃𝑋𝑃 ∈ {𝑃})
2827ad2antlr 765 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29 isopn3i 21088 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3029adantlr 753 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3128, 30eleqtrrd 2842 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
3226, 31impbida 913 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
3332notbid 307 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
3416, 33bitrd 268 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
359, 13, 343bitrd 294 . . 3 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
3635pm5.32da 676 . 2 (𝐽 ∈ Top → ((𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
376, 36bitrd 268 1 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  cdif 3712  wss 3715  {csn 4321   cuni 4588  cfv 6049  Topctop 20900  intcnt 21023  clsccl 21024  limPtclp 21140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-ntr 21026  df-cls 21027  df-lp 21142
This theorem is referenced by:  isperf3  21159
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