Theorem maxlp 23050

Description: A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
lpfval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
maxlp	⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))

Proof of Theorem maxlp

Step	Hyp	Ref	Expression
1		ssid 3960	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
2		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	lpss 23045	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
4	1, 3	mpan2 691	. . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
5	4	sseld 3936	. . 3 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃 ∈ 𝑋))
6	5	pm4.71rd 562	. 2 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7		simpl 482	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ Top)
8	2	islp 23043	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
9	7, 1, 8	sylancl 586	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10		snssi 4762	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → {𝑃} ⊆ 𝑋)
11	2	clsdif 22956	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
12	10, 11	sylan2 593	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
13	12	eleq2d 2814	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14		eldif 3915	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
15	14	baib 535	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
16	15	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17		snssi 4762	. . . . . . . . . 10 ⊢ (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
18	17	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
19	2	ntrss2 22960	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
20	10, 19	sylan2 593	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
21	20	adantr 480	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
22	18, 21	eqssd 3955	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
23	2	ntropn 22952	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
24	10, 23	sylan2 593	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
25	24	adantr 480	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
26	22, 25	eqeltrd 2828	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27		snidg 4614	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑋 → 𝑃 ∈ {𝑃})
28	27	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29		isopn3i 22985	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
30	29	adantlr 715	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
31	28, 30	eleqtrrd 2831	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
32	26, 31	impbida 800	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
33	32	notbid 318	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
34	16, 33	bitrd 279	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
35	9, 13, 34	3bitrd 305	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
36	35	pm5.32da 579	. 2 ⊢ (𝐽 ∈ Top → ((𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
37	6, 36	bitrd 279	1 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3902   ⊆ wss 3905  {csn 4579  ∪ cuni 4861  ‘cfv 6486  Topctop 22796  intcnt 22920  clsccl 22921  limPtclp 23037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-top 22797  df-cld 22922  df-ntr 22923  df-cls 22924  df-lp 23039

This theorem is referenced by:  isperf3  23056