Theorem maxlp 23285

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 23280	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
4	1, 3	mpan2 703	. . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
5	4	sseld 3937	. . 3 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) → 𝑃 ∈ 𝑋))
6	5	pm4.71rd 571	. 2 ⊢ (𝐽 ∈ Top → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑃 ∈ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑋))))
7		simpl 487	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ Top)
8	2	islp 23278	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
9	7, 1, 8	sylancl 597	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10		snssi 4752	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → {𝑃} ⊆ 𝑋)
11	2	clsdif 23191	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
12	10, 11	sylan2 604	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
13	12	eleq2d 2849	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14		eldif 3916	. . . . . . 7 ⊢ (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ (𝑃 ∈ 𝑋 ∧ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
15	14	baib 544	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
16	15	adantl 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃})) ↔ ¬ 𝑃 ∈ ((int‘𝐽)‘{𝑃})))
17		snssi 4752	. . . . . . . . . 10 ⊢ (𝑃 ∈ ((int‘𝐽)‘{𝑃}) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
18	17	adantl 486	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ⊆ ((int‘𝐽)‘{𝑃}))
19	2	ntrss2 23195	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
20	10, 19	sylan2 604	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
21	20	adantr 485	. . . . . . . . 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 23187	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
24	10, 23	sylan2 604	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
25	24	adantr 485	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
26	22, 25	eqeltrd 2863	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27		snidg 4627	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑋 → 𝑃 ∈ {𝑃})
28	27	ad2antlr 739	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29		isopn3i 23220	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
30	29	adantlr 727	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
31	28, 30	eleqtrrd 2866	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
32	26, 31	impbida 812	. . . . . 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 589	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ⊆ wss 3906  {csn 4590  ∪ cuni 4873  ‘cfv 6538  Topctop 23031  intcnt 23155  clsccl 23156  limPtclp 23272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 23032  df-cld 23157  df-ntr 23158  df-cls 23159  df-lp 23274

This theorem is referenced by:  isperf3  23291