Theorem seposep 49860

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 49858. The relationship between separatedness and closure is also seen in isnrm 23566, isnrm2 23589, isnrm3 23590. (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 2762	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	eltopss 23138	. . . . . . 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 3944	. . . . 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 23138	. . . . . . 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 3944	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
15	6	opncld 23264	. . . . . . . . . 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 4158	. . . . . . . . . . . 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 2811	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
20		reldisj 4409	. . . . . . . . . . . 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 3944	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛))
24	6	clsss2 23303	. . . . . . . . 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 4096	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ⊆ (∪ 𝐽 ∖ 𝑆))
27	25, 26	sstrd 3944	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑆))
28		disjdif 4429	. . . . . . . 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 49745	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
31	6	opncld 23264	. . . . . . . . . 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 4409	. . . . . . . . . . . 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 3944	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚))
37	6	clsss2 23303	. . . . . . . . 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 4096	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ⊆ (∪ 𝐽 ∖ 𝑇))
40	38, 39	sstrd 3944	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑇))
41		disjdifr 4430	. . . . . . . 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 49744	. . . . . 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 3220	. 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 2145  ∃wrex 3088   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ‘cfv 6537  Topctop 23124  Clsdccld 23247  clsccl 23249

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23125  df-cld 23250  df-cls 23252

This theorem is used by: (None)