Theorem maxlp 23470

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 3930	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
2		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	lpss 23465	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
4	1, 3	mpan2 704	. . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
5	4	sseld 3907	. . 3 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃 ∈ 𝑋))
6	5	pm4.71rd 572	. 2 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7		simpl 488	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ Top)
8	2	islp 23463	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
9	7, 1, 8	sylancl 598	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10		snssi 4742	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → {𝑃} ⊆ 𝑋)
11	2	clsdif 23376	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
12	10, 11	sylan2 605	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
13	12	eleq2d 2846	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14		eldif 4049	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
15	14	baib 545	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
16	15	adantl 487	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17		snssi 4742	. . . . . . . . . 10 ⊢ (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
18	17	adantl 487	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
19	2	ntrss2 23380	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
20	10, 19	sylan2 605	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
21	20	adantr 486	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
22	18, 21	eqssd 3925	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
23	2	ntropn 23372	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
24	10, 23	sylan2 605	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
25	24	adantr 486	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
26	22, 25	eqeltrd 2860	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27		snidg 4617	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑋 → 𝑃 ∈ {𝑃})
28	27	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29		isopn3i 23405	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
30	29	adantlr 728	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
31	28, 30	eleqtrrd 2863	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
32	26, 31	impbida 813	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ {𝑃} ∈ 𝐽))
33	32	notbid 321	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃}) ↔ ¬ {𝑃} ∈ 𝐽))
34	16, 33	bitrd 282	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ {𝑃} ∈ 𝐽))
35	9, 13, 34	3bitrd 308	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑃} ∈ 𝐽))
36	35	pm5.32da 590	. 2 ⊢ (𝐽 ∈ Top → ((𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑋)) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))
37	6, 36	bitrd 282	1 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ ¬ {𝑃} ∈ 𝐽)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∖ cdif 4045  {csn 4584  ∪ cuni 4866  ‘cfv 6538  Topctop 23216  intcnt 23340  clsccl 23341  limPtclp 23457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23217  df-cld 23342  df-ntr 23343  df-cls 23344  df-lp 23459

This theorem is used by:  isperf3  23476