Theorem seposep 48823

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 48821. The relationship between separatedness and closure is also seen in isnrm 23343, isnrm2 23366, isnrm3 23367. (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 1210	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ 𝑛)
4		simp1 1137	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐽 ∈ Top)
5		simp2l 1200	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ∈ 𝐽)
6		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	eltopss 22913	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ ∪ 𝐽)
9	3, 8	sstrd 3994	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ ∪ 𝐽)
10		simp32 1211	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ 𝑚)
11		simp2r 1201	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ∈ 𝐽)
12	6	eltopss 22913	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → 𝑚 ⊆ ∪ 𝐽)
13	4, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ ∪ 𝐽)
14	10, 13	sstrd 3994	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
15	6	opncld 23041	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
16	4, 5, 15	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
17		incom 4209	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ 𝑚) = (𝑚 ∩ 𝑛)
18		simp33 1212	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑚) = ∅)
19	17, 18	eqtr3id 2791	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
20		reldisj 4453	. . . . . . . . . . . 12 ⊢ (𝑚 ⊆ ∪ 𝐽 → ((𝑚 ∩ 𝑛) = ∅ ↔ 𝑚 ⊆ (∪ 𝐽 ∖ 𝑛)))
21	20	biimpd 229	. . . . . . . . . . 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 3994	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛))
24	6	clsss2 23080	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
25	16, 23, 24	syl2anc 584	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
26	3	sscond 4146	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ⊆ (∪ 𝐽 ∖ 𝑆))
27	25, 26	sstrd 3994	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑆))
28		disjdif 4472	. . . . . . . 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 48728	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
31	6	opncld 23041	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
32	4, 11, 31	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
33		reldisj 4453	. . . . . . . . . . . 12 ⊢ (𝑛 ⊆ ∪ 𝐽 → ((𝑛 ∩ 𝑚) = ∅ ↔ 𝑛 ⊆ (∪ 𝐽 ∖ 𝑚)))
34	33	biimpd 229	. . . . . . . . . . 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 3994	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚))
37	6	clsss2 23080	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
38	32, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
39	10	sscond 4146	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ⊆ (∪ 𝐽 ∖ 𝑇))
40	38, 39	sstrd 3994	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑇))
41		disjdifr 4473	. . . . . . . 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 48727	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
44	30, 43	jca 511	. . . . 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 514	. . . 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 1120	. . 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 3212	. 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 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029

This theorem is referenced by: (None)