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Theorem maxlp 21277
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 3817 . . . . 5 𝑋𝑋
2 lpfval.1 . . . . . 6 𝑋 = 𝐽
32lpss 21272 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
41, 3mpan2 683 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
54sseld 3795 . . 3 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃𝑋))
65pm4.71rd 559 . 2 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7 simpl 475 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → 𝐽 ∈ Top)
82islp 21270 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
97, 1, 8sylancl 581 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10 snssi 4525 . . . . . 6 (𝑃𝑋 → {𝑃} ⊆ 𝑋)
112clsdif 21183 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1210, 11sylan2 587 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1312eleq2d 2862 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14 eldif 3777 . . . . . . 7 (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1514baib 532 . . . . . 6 (𝑃𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
1615adantl 474 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17 snssi 4525 . . . . . . . . . 10 (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
1817adantl 474 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
192ntrss2 21187 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2010, 19sylan2 587 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2120adantr 473 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2218, 21eqssd 3813 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
232ntropn 21179 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2410, 23sylan2 587 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2524adantr 473 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2876 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27 snidg 4396 . . . . . . . . 9 (𝑃𝑋𝑃 ∈ {𝑃})
2827ad2antlr 719 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29 isopn3i 21212 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3029adantlr 707 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3128, 30eleqtrrd 2879 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
3226, 31impbida 836 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
3332notbid 310 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
3416, 33bitrd 271 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
359, 13, 343bitrd 297 . . 3 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
3635pm5.32da 575 . 2 (𝐽 ∈ Top → ((𝑃𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
376, 36bitrd 271 1 (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cdif 3764  wss 3767  {csn 4366   cuni 4626  cfv 6099  Topctop 21023  intcnt 21147  clsccl 21148  limPtclp 21264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-top 21024  df-cld 21149  df-ntr 21150  df-cls 21151  df-lp 21266
This theorem is referenced by:  isperf3  21283
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