Theorem seposep 49745

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 49743. The relationship between separatedness and closure is also seen in isnrm 23529, isnrm2 23552, isnrm3 23553. (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 1228	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ 𝑛)
4		simp1 1154	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐽 ∈ Top)
5		simp2l 1218	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ∈ 𝐽)
6		eqid 2766	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	eltopss 23101	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 596	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ ∪ 𝐽)
9	3, 8	sstrd 3950	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ ∪ 𝐽)
10		simp32 1229	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ 𝑚)
11		simp2r 1219	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ∈ 𝐽)
12	6	eltopss 23101	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → 𝑚 ⊆ ∪ 𝐽)
13	4, 11, 12	syl2anc 596	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ ∪ 𝐽)
14	10, 13	sstrd 3950	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
15	6	opncld 23227	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
16	4, 5, 15	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
17		incom 4165	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ 𝑚) = (𝑚 ∩ 𝑛)
18		simp33 1230	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑚) = ∅)
19	17, 18	eqtr3id 2815	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
20		reldisj 4416	. . . . . . . . . . . 12 ⊢ (𝑚 ⊆ ∪ 𝐽 → ((𝑚 ∩ 𝑛) = ∅ ↔ 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛)))
21	20	biimpd 232	. . . . . . . . . . 11 ⊢ (𝑚 ⊆ ∪ 𝐽 → ((𝑚 ∩ 𝑛) = ∅ → 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛)))
22	13, 19, 21	sylc 66	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛))
23	10, 22	sstrd 3950	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛))
24	6	clsss2 23266	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
25	16, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
26	3	sscond 4103	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ⊆ (∪ 𝐽 ∖ 𝑆))
27	25, 26	sstrd 3950	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑆))
28		disjdif 4436	. . . . . . . 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 49628	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
31	6	opncld 23227	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
32	4, 11, 31	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
33		reldisj 4416	. . . . . . . . . . . 12 ⊢ (𝑛 ⊆ ∪ 𝐽 → ((𝑛 ∩ 𝑚) = ∅ ↔ 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚)))
34	33	biimpd 232	. . . . . . . . . . 11 ⊢ (𝑛 ⊆ ∪ 𝐽 → ((𝑛 ∩ 𝑚) = ∅ → 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚)))
35	8, 18, 34	sylc 66	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚))
36	3, 35	sstrd 3950	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚))
37	6	clsss2 23266	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
38	32, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
39	10	sscond 4103	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ⊆ (∪ 𝐽 ∖ 𝑇))
40	38, 39	sstrd 3950	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑇))
41		disjdifr 4437	. . . . . . . 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 49627	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
44	30, 43	jca 521	. . . . 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 524	. . . 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 1137	. . 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 3224	. 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 66	1 ⊢ (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  ∪ cuni 4877  ‘cfv 6543  Topctop 23087  Clsdccld 23210  clsccl 23212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  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 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 23088  df-cld 23213  df-cls 23215

This theorem is used by: (None)