Theorem maxlp 22973

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 3996	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
2		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	lpss 22968	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
4	1, 3	mpan2 688	. . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
5	4	sseld 3973	. . 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 22966	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
9	7, 1, 8	sylancl 585	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10		snssi 4803	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → {𝑃} ⊆ 𝑋)
11	2	clsdif 22879	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
12	10, 11	sylan2 592	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
13	12	eleq2d 2811	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14		eldif 3950	. . . . . . 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 4803	. . . . . . . . . 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 22883	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
20	10, 19	sylan2 592	. . . . . . . . . 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 3991	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
23	2	ntropn 22875	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
24	10, 23	sylan2 592	. . . . . . . . 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 2825	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27		snidg 4654	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑋 → 𝑃 ∈ {𝑃})
28	27	ad2antlr 724	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29		isopn3i 22908	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
30	29	adantlr 712	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
31	28, 30	eleqtrrd 2828	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
32	26, 31	impbida 798	. . . . . 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 578	. 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 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ∖ cdif 3937   ⊆ wss 3940  {csn 4620  ∪ cuni 4899  ‘cfv 6533  Topctop 22717  intcnt 22843  clsccl 22844  limPtclp 22960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 22718  df-cld 22845  df-ntr 22846  df-cls 22847  df-lp 22962

This theorem is referenced by:  isperf3  22979