Theorem maxlp 23103

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 3958	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
2		lpfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	lpss 23098	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
4	1, 3	mpan2 692	. . . 4 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑋) ⊆ 𝑋)
5	4	sseld 3934	. . 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 23096	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
9	7, 1, 8	sylancl 587	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑋) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃}))))
10		snssi 4766	. . . . . 6 ⊢ (𝑃 ∈ 𝑋 → {𝑃} ⊆ 𝑋)
11	2	clsdif 23009	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
12	10, 11	sylan2 594	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) = (𝑋 ∖ ((int‘𝐽)‘{𝑃})))
13	12	eleq2d 2823	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑋 ∖ {𝑃})) ↔ 𝑃 ∈ (𝑋 ∖ ((int‘𝐽)‘{𝑃}))))
14		eldif 3913	. . . . . . 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 4766	. . . . . . . . . 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 23013	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ⊆ {𝑃})
20	10, 19	sylan2 594	. . . . . . . . . 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 3953	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} = ((int‘𝐽)‘{𝑃}))
23	2	ntropn 23005	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ⊆ 𝑋) → ((int‘𝐽)‘{𝑃}) ∈ 𝐽)
24	10, 23	sylan2 594	. . . . . . . . 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 2837	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ 𝑃 ∈ ((int‘𝐽)‘{𝑃})) → {𝑃} ∈ 𝐽)
27		snidg 4619	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑋 → 𝑃 ∈ {𝑃})
28	27	ad2antlr 728	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ {𝑃})
29		isopn3i 23038	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
30	29	adantlr 716	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → ((int‘𝐽)‘{𝑃}) = {𝑃})
31	28, 30	eleqtrrd 2840	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ {𝑃} ∈ 𝐽) → 𝑃 ∈ ((int‘𝐽)‘{𝑃}))
32	26, 31	impbida 801	. . . . . 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 1542   ∈ wcel 2114   ∖ cdif 3900   ⊆ wss 3903  {csn 4582  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  intcnt 22973  clsccl 22974  limPtclp 23090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977  df-lp 23092

This theorem is referenced by:  isperf3  23109