Theorem seposep 48605

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 48603. The relationship between separatedness and closure is also seen in isnrm 23364, isnrm2 23387, isnrm3 23388. (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 1209	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ 𝑛)
4		simp1 1136	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝐽 ∈ Top)
5		simp2l 1199	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ∈ 𝐽)
6		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	eltopss 22934	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 583	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑛 ⊆ ∪ 𝐽)
9	3, 8	sstrd 4019	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ ∪ 𝐽)
10		simp32 1210	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ 𝑚)
11		simp2r 1200	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ∈ 𝐽)
12	6	eltopss 22934	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → 𝑚 ⊆ ∪ 𝐽)
13	4, 11, 12	syl2anc 583	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑚 ⊆ ∪ 𝐽)
14	10, 13	sstrd 4019	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ ∪ 𝐽)
15	6	opncld 23062	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
16	4, 5, 15	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽))
17		incom 4230	. . . . . . . . . . . 12 ⊢ (𝑛 ∩ 𝑚) = (𝑚 ∩ 𝑛)
18		simp33 1211	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑛 ∩ 𝑚) = ∅)
19	17, 18	eqtr3id 2794	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑚 ∩ 𝑛) = ∅)
20		reldisj 4476	. . . . . . . . . . . 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 4019	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛))
24	6	clsss2 23101	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑛) ∈ (Clsd‘𝐽) ∧ 𝑇 ⊆ (∪ 𝐽 ∖ 𝑛)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
25	16, 23, 24	syl2anc 583	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑛))
26	3	sscond 4169	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑛) ⊆ (∪ 𝐽 ∖ 𝑆))
27	25, 26	sstrd 4019	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑇) ⊆ (∪ 𝐽 ∖ 𝑆))
28		disjdif 4495	. . . . . . . 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 48540	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅)
31	6	opncld 23062	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑚 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
32	4, 11, 31	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽))
33		reldisj 4476	. . . . . . . . . . . 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 4019	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚))
37	6	clsss2 23101	. . . . . . . . 9 ⊢ (((∪ 𝐽 ∖ 𝑚) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (∪ 𝐽 ∖ 𝑚)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
38	32, 36, 37	syl2anc 583	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑚))
39	10	sscond 4169	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → (∪ 𝐽 ∖ 𝑚) ⊆ (∪ 𝐽 ∖ 𝑇))
40	38, 39	sstrd 4019	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑛 ∈ 𝐽 ∧ 𝑚 ∈ 𝐽) ∧ (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((cls‘𝐽)‘𝑆) ⊆ (∪ 𝐽 ∖ 𝑇))
41		disjdifr 4496	. . . . . . . 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 48539	. . . . . 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 1119	. . 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 3218	. 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 1537   ∈ wcel 2108  ∃wrex 3076   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  Clsdccld 23045  clsccl 23047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-cld 23048  df-cls 23050

This theorem is referenced by: (None)