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Theorem islpi 22873
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 = βˆͺ 𝐽
Assertion
Ref Expression
islpi (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ Β¬ 𝑃 ∈ 𝑆)) β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†))
Proof of Theorem islpi
StepHypRef Expression
1 lpfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
21clslp 22872 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
32eleq2d 2817 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ 𝑃 ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))))
4 elun 4147 . . . . 5 (𝑃 ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†)))
5 df-or 844 . . . . 5 ((𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†)) ↔ (Β¬ 𝑃 ∈ 𝑆 β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†)))
64, 5bitri 274 . . . 4 (𝑃 ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) ↔ (Β¬ 𝑃 ∈ 𝑆 β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†)))
73, 6bitrdi 286 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ (Β¬ 𝑃 ∈ 𝑆 β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†))))
87biimpd 228 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ (Β¬ 𝑃 ∈ 𝑆 β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†))))
98imp32 417 1 (((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) ∧ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ Β¬ 𝑃 ∈ 𝑆)) β†’ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   βˆͺ cun 3945   βŠ† wss 3947  βˆͺ cuni 4907  β€˜cfv 6542  Topctop 22615  clsccl 22742  limPtclp 22858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22616  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lp 22860
This theorem is referenced by: (None)
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