Theorem seposep 49423

Description: If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 49421. The relationship between separatedness and closure is also seen in isnrm 23325, isnrm2 23348, isnrm3 23349. (Contributed by Zhi Wang, 7-Sep-2024.)

Hypotheses

Ref	Expression
sepdisj.1	⊢ (𝜑 → 𝐽 ∈ Top)
seposep.2	⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))

Assertion

Ref	Expression
seposep	⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))

Distinct variable groups:   𝑚,𝐽,𝑛   𝑆,𝑚,𝑛   𝑇,𝑚,𝑛

Allowed substitution hints:   𝜑(𝑚,𝑛)

Proof of Theorem seposep

Step	Hyp	Ref	Expression
1		sepdisj.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		seposep.2	. 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
3		simp31 1216	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ 𝑛)
4		simp1 1142	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐽 ∈ Top)
5		simp2l 1206	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ∈ 𝐽)
6		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	eltopss 22897	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 590	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ ∪ 𝐽)
9	3, 8	sstrd 3932	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ ∪ 𝐽)
10		simp32 1217	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ 𝑚)
11		simp2r 1207	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ∈ 𝐽)
12	6	eltopss 22897	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → 𝑚 ⊆ ∪ 𝐽)
13	4, 11, 12	syl2anc 590	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ ∪ 𝐽)
14	10, 13	sstrd 3932	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
15	6	opncld 23023	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
16	4, 5, 15	syl2anc 590	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
17		incom 4145	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ 𝑚) = (𝑚 ∩ 𝑛)
18		simp33 1218	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑚) = ∅)
19	17, 18	eqtr3id 2789	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
20		reldisj 4388	. . . . . . . . . . . 12 ⊢ (𝑚 ⊆ ∪ 𝐽 → ((𝑚 ∩ 𝑛) = ∅ ↔ 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛)))
21	20	biimpd 230	. . . . . . . . . . 11 ⊢ (𝑚 ⊆ ∪ 𝐽 → ((𝑚 ∩ 𝑛) = ∅ → 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛)))
22	13, 19, 21	sylc 65	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛))
23	10, 22	sstrd 3932	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛))
24	6	clsss2 23062	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
25	16, 23, 24	syl2anc 590	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
26	3	sscond 4083	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ⊆ (∪ 𝐽 ∖ 𝑆))
27	25, 26	sstrd 3932	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑆))
28		disjdif 4407	. . . . . . . 8 ⊢ (𝑆 ∩ (∪ 𝐽 ∖ 𝑆)) = ∅
29	28	a1i 11	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ (∪ 𝐽 ∖ 𝑆)) = ∅)
30	27, 29	ssdisjdr 49306	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
31	6	opncld 23023	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
32	4, 11, 31	syl2anc 590	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
33		reldisj 4388	. . . . . . . . . . . 12 ⊢ (𝑛 ⊆ ∪ 𝐽 → ((𝑛 ∩ 𝑚) = ∅ ↔ 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚)))
34	33	biimpd 230	. . . . . . . . . . 11 ⊢ (𝑛 ⊆ ∪ 𝐽 → ((𝑛 ∩ 𝑚) = ∅ → 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚)))
35	8, 18, 34	sylc 65	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚))
36	3, 35	sstrd 3932	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚))
37	6	clsss2 23062	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
38	32, 36, 37	syl2anc 590	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
39	10	sscond 4083	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ⊆ (∪ 𝐽 ∖ 𝑇))
40	38, 39	sstrd 3932	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑇))
41		disjdifr 4408	. . . . . . . 8 ⊢ ((∪ 𝐽 ∖ 𝑇) ∩ 𝑇) = ∅
42	41	a1i 11	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((∪ 𝐽 ∖ 𝑇) ∩ 𝑇) = ∅)
43	40, 42	ssdisjd 49305	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
44	30, 43	jca 516	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
45	9, 14, 44	jca31 519	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
46	45	3exp 1125	. . 3 ⊢ (𝐽 ∈ Top → ((𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) → ((𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))))
47	46	rexlimdvv 3196	. 2 ⊢ (𝐽 ∈ Top → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))))
48	1, 2, 47	sylc 65	1 ⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  ∪ cuni 4845  ‘cfv 6492  Topctop 22883  Clsdccld 23006  clsccl 23008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22884  df-cld 23009  df-cls 23011

This theorem is referenced by: (None)