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Theorem islpi 22579
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 22578 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘†) = (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)))
32eleq2d 2818 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑃 ∈ ((clsβ€˜π½)β€˜π‘†) ↔ 𝑃 ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†))))
4 elun 4143 . . . . 5 (𝑃 ∈ (𝑆 βˆͺ ((limPtβ€˜π½)β€˜π‘†)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPtβ€˜π½)β€˜π‘†)))
5 df-or 846 . . . . 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 419 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   βˆͺ cun 3941   βŠ† wss 3943  βˆͺ cuni 4900  β€˜cfv 6531  Topctop 22321  clsccl 22448  limPtclp 22564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5277  ax-sep 5291  ax-nul 5298  ax-pow 5355  ax-pr 5419  ax-un 7707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3474  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4943  df-iun 4991  df-iin 4992  df-br 5141  df-opab 5203  df-mpt 5224  df-id 5566  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-iota 6483  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-top 22322  df-cld 22449  df-ntr 22450  df-cls 22451  df-nei 22528  df-lp 22566
This theorem is referenced by: (None)
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