Theorem islpi 21754

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 21753	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
3	2	eleq2d 2875	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4		elun 4076	. . . . 5 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5		df-or 845	. . . . 5 ⊢ ((𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
6	4, 5	bitri 278	. . . 4 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
7	3, 6	syl6bb 290	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
8	7	biimpd 232	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) → (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
9	8	imp32 422	1 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃 ∈ 𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  clsccl 21623  limPtclp 21739

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-lp 21741

This theorem is referenced by: (None)