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Theorem islpi 21279
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 21278 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
32eleq2d 2862 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4 elun 3949 . . . . 5 (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5 df-or 875 . . . . 5 ((𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
64, 5bitri 267 . . . 4 (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
73, 6syl6bb 279 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
87biimpd 221 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → (¬ 𝑃𝑆𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
98imp32 410 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  wa 385  wo 874   = wceq 1653  wcel 2157  cun 3765  wss 3767   cuni 4626  cfv 6099  Topctop 21023  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-nei 21228  df-lp 21266
This theorem is referenced by: (None)
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