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Theorem islpi 20763
 Description: A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
islpi (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Proof of Theorem islpi
StepHypRef Expression
1 lpfval.1 . . . . . 6 𝑋 = 𝐽
21clslp 20762 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
32eleq2d 2673 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4 elun 3715 . . . . 5 (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5 df-or 384 . . . . 5 ((𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
64, 5bitri 263 . . . 4 (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
73, 6syl6bb 275 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
87biimpd 218 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
98imp32 448 1 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  clsccl 20632  limPtclp 20748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750 This theorem is referenced by: (None)
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