Theorem islpi 23189

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 23188	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))
3	2	eleq2d 2847	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ 𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))))
4		elun 4106	. . . . 5 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
5		df-or 859	. . . . 5 ⊢ ((𝑃 ∈ 𝑆 ∨ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
6	4, 5	bitri 277	. . . 4 ⊢ (𝑃 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆)))
7	3, 6	bitrdi 289	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ (¬ 𝑃 ∈ 𝑆 → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))))
8	7	biimpd 231	. 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 858   = wceq 1559   ∈ wcel 2141   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4864  ‘cfv 6517  Topctop 22933  clsccl 23058  limPtclp 23174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-top 22934  df-cld 23059  df-ntr 23060  df-cls 23061  df-nei 23138  df-lp 23176

This theorem is referenced by: (None)