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Theorem maxlp 23170
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 4017 . . . . 5 𝑋𝑋
2 lpfval.1 . . . . . 6 𝑋 = 𝐽
32lpss 23165 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
41, 3mpan2 691 . . . 4 (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
54sseld 3993 . . 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 23163 . . . . 5 ((𝐽 ∈ Top ∧ 𝑋𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
97, 1, 8sylancl 586 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10 snssi 4812 . . . . . 6 (𝑃𝑋 → {𝑃} ⊆ 𝑋)
112clsdif 23076 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1210, 11sylan2 593 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
1312eleq2d 2824 . . . 4 ((𝐽 ∈ Top ∧ 𝑃𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14 eldif 3972 . . . . . . 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 4812 . . . . . . . . . 10 (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
1817adantl 481 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
192ntrss2 23080 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2010, 19sylan2 593 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2120adantr 480 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
2218, 21eqssd 4012 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
232ntropn 23072 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2410, 23sylan2 593 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑃𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2524adantr 480 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
2622, 25eqeltrd 2838 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27 snidg 4664 . . . . . . . . 9 (𝑃𝑋𝑃 ∈ {𝑃})
2827ad2antlr 727 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29 isopn3i 23105 . . . . . . . . 9 ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3029adantlr 715 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
3128, 30eleqtrrd 2841 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑃𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
3226, 31impbida 801 . . . . . 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 579 . 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 206  wa 395   = wceq 1536  wcel 2105  cdif 3959  wss 3962  {csn 4630   cuni 4911  cfv 6562  Topctop 22914  intcnt 23040  clsccl 23041  limPtclp 23157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-cld 23042  df-ntr 23043  df-cls 23044  df-lp 23159
This theorem is referenced by:  isperf3  23176
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