Theorem islpi 23114

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 23113	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
3	2	eleq2d 2822	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4		elun 4093	. . . . 5 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5		df-or 849	. . . . 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 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3887   ⊆ wss 3889  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  clsccl 22983  limPtclp 23099

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-top 22859  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-lp 23101

This theorem is referenced by: (None)