Theorem islpi 23178

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 ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃 ∈ 𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Proof of Theorem islpi

Step	Hyp	Ref	Expression
1		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	clslp 23177	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
3	2	eleq2d 2830	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4		elun 4176	. . . . 5 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5		df-or 847	. . . . 5 ⊢ ((𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
6	4, 5	bitri 275	. . . 4 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
7	3, 6	bitrdi 287	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
8	7	biimpd 229	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
9	8	imp32 418	1 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃 ∈ 𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  clsccl 23047  limPtclp 23163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-lp 23165

This theorem is referenced by: (None)