sepfsepc - Mathbox for Zhi Wang

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Theorem sepfsepc 48210

Description: If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)

**Hypothesis**
Ref	Expression
sepfsepc.1	⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1})))

**Assertion**
Ref	Expression
sepfsepc	⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))

Distinct variable groups: 𝑓,𝐽,𝑚,𝑛 𝑆,𝑓,𝑛 𝑇,𝑓,𝑚,𝑛 Allowed substitution hints: 𝜑(𝑓,𝑚,𝑛) 𝑆(𝑚)

**Proof of Theorem sepfsepc**
Step	Hyp	Ref	Expression
1		sepfsepc.1	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1})))
2		simpl 481	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → 𝑓 ∈ (𝐽 Cn II))
3		0re 11262	. . . . . . . 8 ⊢ 0 ∈ ℝ
4		1re 11260	. . . . . . . 8 ⊢ 1 ∈ ℝ
5		0le0 12360	. . . . . . . 8 ⊢ 0 ≤ 0
6		3re 12339	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
7		3ne0 12365	. . . . . . . . . 10 ⊢ 3 ≠ 0
8	6, 7	rereccli 12026	. . . . . . . . 9 ⊢ (1 / 3) ∈ ℝ
9		1lt3 12432	. . . . . . . . . . 11 ⊢ 1 < 3
10		recgt1i 12158	. . . . . . . . . . 11 ⊢ ((3 ∈ ℝ ∧ 1 < 3) → (0 < (1 / 3) ∧ (1 / 3) < 1))
11	6, 9, 10	mp2an 690	. . . . . . . . . 10 ⊢ (0 < (1 / 3) ∧ (1 / 3) < 1)
12	11	simpri 484	. . . . . . . . 9 ⊢ (1 / 3) < 1
13	8, 4, 12	ltleii 11383	. . . . . . . 8 ⊢ (1 / 3) ≤ 1
14		iccss 13441	. . . . . . . 8 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ 0 ∧ (1 / 3) ≤ 1)) → (0[,](1 / 3)) ⊆ (0[,]1))
15	3, 4, 5, 13, 14	mp4an 691	. . . . . . 7 ⊢ (0[,](1 / 3)) ⊆ (0[,]1)
16		i0oii 48202	. . . . . . . . 9 ⊢ ((1 / 3) ≤ 1 → (0[,)(1 / 3)) ∈ II)
17	13, 16	ax-mp 5	. . . . . . . 8 ⊢ (0[,)(1 / 3)) ∈ II
18	11	simpli 482	. . . . . . . . . 10 ⊢ 0 < (1 / 3)
19	8	rexri 11318	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ^*
20		elico2 13437	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (1 / 3) ∈ ℝ^*) → (0 ∈ (0[,)(1 / 3)) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 < (1 / 3))))
21	3, 19, 20	mp2an 690	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,)(1 / 3)) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 < (1 / 3)))
22	21	biimpri 227	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 < (1 / 3)) → 0 ∈ (0[,)(1 / 3)))
23	22	snssd 4817	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 < (1 / 3)) → {0} ⊆ (0[,)(1 / 3)))
24	3, 5, 18, 23	mp3an 1457	. . . . . . . . 9 ⊢ {0} ⊆ (0[,)(1 / 3))
25		icossicc 13462	. . . . . . . . 9 ⊢ (0[,)(1 / 3)) ⊆ (0[,](1 / 3))
26	24, 25	pm3.2i 469	. . . . . . . 8 ⊢ ({0} ⊆ (0[,)(1 / 3)) ∧ (0[,)(1 / 3)) ⊆ (0[,](1 / 3)))
27		sseq2 4005	. . . . . . . . . 10 ⊢ (𝑔 = (0[,)(1 / 3)) → ({0} ⊆ 𝑔 ↔ {0} ⊆ (0[,)(1 / 3))))
28		sseq1 4004	. . . . . . . . . 10 ⊢ (𝑔 = (0[,)(1 / 3)) → (𝑔 ⊆ (0[,](1 / 3)) ↔ (0[,)(1 / 3)) ⊆ (0[,](1 / 3))))
29	27, 28	anbi12d 630	. . . . . . . . 9 ⊢ (𝑔 = (0[,)(1 / 3)) → (({0} ⊆ 𝑔 ∧ 𝑔 ⊆ (0[,](1 / 3))) ↔ ({0} ⊆ (0[,)(1 / 3)) ∧ (0[,)(1 / 3)) ⊆ (0[,](1 / 3)))))
30	29	rspcev 3607	. . . . . . . 8 ⊢ (((0[,)(1 / 3)) ∈ II ∧ ({0} ⊆ (0[,)(1 / 3)) ∧ (0[,)(1 / 3)) ⊆ (0[,](1 / 3)))) → ∃𝑔 ∈ II ({0} ⊆ 𝑔 ∧ 𝑔 ⊆ (0[,](1 / 3))))
31	17, 26, 30	mp2an 690	. . . . . . 7 ⊢ ∃𝑔 ∈ II ({0} ⊆ 𝑔 ∧ 𝑔 ⊆ (0[,](1 / 3)))
32		iitop 24883	. . . . . . . 8 ⊢ II ∈ Top
33	24, 25	sstri 3988	. . . . . . . . 9 ⊢ {0} ⊆ (0[,](1 / 3))
34	33, 15	sstri 3988	. . . . . . . 8 ⊢ {0} ⊆ (0[,]1)
35		iiuni 24884	. . . . . . . . 9 ⊢ (0[,]1) = ∪ II
36	35	isnei 23090	. . . . . . . 8 ⊢ ((II ∈ Top ∧ {0} ⊆ (0[,]1)) → ((0[,](1 / 3)) ∈ ((nei‘II)‘{0}) ↔ ((0[,](1 / 3)) ⊆ (0[,]1) ∧ ∃𝑔 ∈ II ({0} ⊆ 𝑔 ∧ 𝑔 ⊆ (0[,](1 / 3))))))
37	32, 34, 36	mp2an 690	. . . . . . 7 ⊢ ((0[,](1 / 3)) ∈ ((nei‘II)‘{0}) ↔ ((0[,](1 / 3)) ⊆ (0[,]1) ∧ ∃𝑔 ∈ II ({0} ⊆ 𝑔 ∧ 𝑔 ⊆ (0[,](1 / 3)))))
38	15, 31, 37	mpbir2an 709	. . . . . 6 ⊢ (0[,](1 / 3)) ∈ ((nei‘II)‘{0})
39	38	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (0[,](1 / 3)) ∈ ((nei‘II)‘{0}))
40		simprl 769	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → 𝑆 ⊆ (^◡𝑓 “ {0}))
41	2, 39, 40	cnneiima 48199	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ (0[,](1 / 3))) ∈ ((nei‘𝐽)‘𝑆))
42		halfge0 12476	. . . . . . . 8 ⊢ 0 ≤ (1 / 2)
43		1le1 11888	. . . . . . . 8 ⊢ 1 ≤ 1
44		iccss 13441	. . . . . . . 8 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ (1 / 2) ∧ 1 ≤ 1)) → ((1 / 2)[,]1) ⊆ (0[,]1))
45	3, 4, 42, 43, 44	mp4an 691	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ (0[,]1)
46		io1ii 48203	. . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → ((1 / 2)(,]1) ∈ II)
47	42, 46	ax-mp 5	. . . . . . . 8 ⊢ ((1 / 2)(,]1) ∈ II
48		halflt1 12477	. . . . . . . . . 10 ⊢ (1 / 2) < 1
49		halfre 12473	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℝ
50	49	rexri 11318	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ^*
51		elioc2 13436	. . . . . . . . . . . . 13 ⊢ (((1 / 2) ∈ ℝ^* ∧ 1 ∈ ℝ) → (1 ∈ ((1 / 2)(,]1) ↔ (1 ∈ ℝ ∧ (1 / 2) < 1 ∧ 1 ≤ 1)))
52	50, 4, 51	mp2an 690	. . . . . . . . . . . 12 ⊢ (1 ∈ ((1 / 2)(,]1) ↔ (1 ∈ ℝ ∧ (1 / 2) < 1 ∧ 1 ≤ 1))
53	52	biimpri 227	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ (1 / 2) < 1 ∧ 1 ≤ 1) → 1 ∈ ((1 / 2)(,]1))
54	53	snssd 4817	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (1 / 2) < 1 ∧ 1 ≤ 1) → {1} ⊆ ((1 / 2)(,]1))
55	4, 48, 43, 54	mp3an 1457	. . . . . . . . 9 ⊢ {1} ⊆ ((1 / 2)(,]1)
56		iocssicc 13463	. . . . . . . . 9 ⊢ ((1 / 2)(,]1) ⊆ ((1 / 2)[,]1)
57	55, 56	pm3.2i 469	. . . . . . . 8 ⊢ ({1} ⊆ ((1 / 2)(,]1) ∧ ((1 / 2)(,]1) ⊆ ((1 / 2)[,]1))
58		sseq2 4005	. . . . . . . . . 10 ⊢ (ℎ = ((1 / 2)(,]1) → ({1} ⊆ ℎ ↔ {1} ⊆ ((1 / 2)(,]1)))
59		sseq1 4004	. . . . . . . . . 10 ⊢ (ℎ = ((1 / 2)(,]1) → (ℎ ⊆ ((1 / 2)[,]1) ↔ ((1 / 2)(,]1) ⊆ ((1 / 2)[,]1)))
60	58, 59	anbi12d 630	. . . . . . . . 9 ⊢ (ℎ = ((1 / 2)(,]1) → (({1} ⊆ ℎ ∧ ℎ ⊆ ((1 / 2)[,]1)) ↔ ({1} ⊆ ((1 / 2)(,]1) ∧ ((1 / 2)(,]1) ⊆ ((1 / 2)[,]1))))
61	60	rspcev 3607	. . . . . . . 8 ⊢ ((((1 / 2)(,]1) ∈ II ∧ ({1} ⊆ ((1 / 2)(,]1) ∧ ((1 / 2)(,]1) ⊆ ((1 / 2)[,]1))) → ∃ℎ ∈ II ({1} ⊆ ℎ ∧ ℎ ⊆ ((1 / 2)[,]1)))
62	47, 57, 61	mp2an 690	. . . . . . 7 ⊢ ∃ℎ ∈ II ({1} ⊆ ℎ ∧ ℎ ⊆ ((1 / 2)[,]1))
63	55, 56	sstri 3988	. . . . . . . . 9 ⊢ {1} ⊆ ((1 / 2)[,]1)
64	63, 45	sstri 3988	. . . . . . . 8 ⊢ {1} ⊆ (0[,]1)
65	35	isnei 23090	. . . . . . . 8 ⊢ ((II ∈ Top ∧ {1} ⊆ (0[,]1)) → (((1 / 2)[,]1) ∈ ((nei‘II)‘{1}) ↔ (((1 / 2)[,]1) ⊆ (0[,]1) ∧ ∃ℎ ∈ II ({1} ⊆ ℎ ∧ ℎ ⊆ ((1 / 2)[,]1)))))
66	32, 64, 65	mp2an 690	. . . . . . 7 ⊢ (((1 / 2)[,]1) ∈ ((nei‘II)‘{1}) ↔ (((1 / 2)[,]1) ⊆ (0[,]1) ∧ ∃ℎ ∈ II ({1} ⊆ ℎ ∧ ℎ ⊆ ((1 / 2)[,]1))))
67	45, 62, 66	mpbir2an 709	. . . . . 6 ⊢ ((1 / 2)[,]1) ∈ ((nei‘II)‘{1})
68	67	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → ((1 / 2)[,]1) ∈ ((nei‘II)‘{1}))
69		simprr 771	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → 𝑇 ⊆ (^◡𝑓 “ {1}))
70	2, 68, 69	cnneiima 48199	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ ((1 / 2)[,]1)) ∈ ((nei‘𝐽)‘𝑇))
71		icccldii 48201	. . . . . 6 ⊢ ((0 ≤ 0 ∧ (1 / 3) ≤ 1) → (0[,](1 / 3)) ∈ (Clsd‘II))
72	5, 13, 71	mp2an 690	. . . . 5 ⊢ (0[,](1 / 3)) ∈ (Clsd‘II)
73		cnclima 23255	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (0[,](1 / 3)) ∈ (Clsd‘II)) → (^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽))
74	2, 72, 73	sylancl 584	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽))
75		icccldii 48201	. . . . . 6 ⊢ ((0 ≤ (1 / 2) ∧ 1 ≤ 1) → ((1 / 2)[,]1) ∈ (Clsd‘II))
76	42, 43, 75	mp2an 690	. . . . 5 ⊢ ((1 / 2)[,]1) ∈ (Clsd‘II)
77		cnclima 23255	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ ((1 / 2)[,]1) ∈ (Clsd‘II)) → (^◡𝑓 “ ((1 / 2)[,]1)) ∈ (Clsd‘𝐽))
78	2, 76, 77	sylancl 584	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ ((1 / 2)[,]1)) ∈ (Clsd‘𝐽))
79		eqid 2725	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
80	79, 35	cnf 23233	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn II) → 𝑓:∪ 𝐽⟶(0[,]1))
81	80	ffund 6731	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn II) → Fun 𝑓)
82	2, 81	syl 17	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → Fun 𝑓)
83		0xr 11307	. . . . . . 7 ⊢ 0 ∈ ℝ^*
84		1xr 11319	. . . . . . 7 ⊢ 1 ∈ ℝ^*
85		2lt3 12431	. . . . . . . 8 ⊢ 2 < 3
86		2re 12333	. . . . . . . . 9 ⊢ 2 ∈ ℝ
87		2pos 12362	. . . . . . . . 9 ⊢ 0 < 2
88		3pos 12364	. . . . . . . . 9 ⊢ 0 < 3
89	86, 6, 87, 88	ltrecii 12177	. . . . . . . 8 ⊢ (2 < 3 ↔ (1 / 3) < (1 / 2))
90	85, 89	mpbi 229	. . . . . . 7 ⊢ (1 / 3) < (1 / 2)
91		iccdisj2 48180	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ (1 / 3) < (1 / 2)) → ((0[,](1 / 3)) ∩ ((1 / 2)[,]1)) = ∅)
92	83, 84, 90, 91	mp3an 1457	. . . . . 6 ⊢ ((0[,](1 / 3)) ∩ ((1 / 2)[,]1)) = ∅
93	92	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → ((0[,](1 / 3)) ∩ ((1 / 2)[,]1)) = ∅)
94		ssidd 4002	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ (0[,](1 / 3))) ⊆ (^◡𝑓 “ (0[,](1 / 3))))
95		ssidd 4002	. . . . 5 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → (^◡𝑓 “ ((1 / 2)[,]1)) ⊆ (^◡𝑓 “ ((1 / 2)[,]1)))
96	82, 93, 94, 95	predisj 48145	. . . 4 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → ((^◡𝑓 “ (0[,](1 / 3))) ∩ (^◡𝑓 “ ((1 / 2)[,]1))) = ∅)
97		eleq1 2813	. . . . . 6 ⊢ (𝑛 = (^◡𝑓 “ (0[,](1 / 3))) → (𝑛 ∈ (Clsd‘𝐽) ↔ (^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽)))
98		ineq1 4205	. . . . . . 7 ⊢ (𝑛 = (^◡𝑓 “ (0[,](1 / 3))) → (𝑛 ∩ 𝑚) = ((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚))
99	98	eqeq1d 2727	. . . . . 6 ⊢ (𝑛 = (^◡𝑓 “ (0[,](1 / 3))) → ((𝑛 ∩ 𝑚) = ∅ ↔ ((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚) = ∅))
100	97, 99	3anbi13d 1434	. . . . 5 ⊢ (𝑛 = (^◡𝑓 “ (0[,](1 / 3))) → ((𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅) ↔ ((^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ ((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚) = ∅)))
101		eleq1 2813	. . . . . 6 ⊢ (𝑚 = (^◡𝑓 “ ((1 / 2)[,]1)) → (𝑚 ∈ (Clsd‘𝐽) ↔ (^◡𝑓 “ ((1 / 2)[,]1)) ∈ (Clsd‘𝐽)))
102		ineq2 4206	. . . . . . 7 ⊢ (𝑚 = (^◡𝑓 “ ((1 / 2)[,]1)) → ((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚) = ((^◡𝑓 “ (0[,](1 / 3))) ∩ (^◡𝑓 “ ((1 / 2)[,]1))))
103	102	eqeq1d 2727	. . . . . 6 ⊢ (𝑚 = (^◡𝑓 “ ((1 / 2)[,]1)) → (((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚) = ∅ ↔ ((^◡𝑓 “ (0[,](1 / 3))) ∩ (^◡𝑓 “ ((1 / 2)[,]1))) = ∅))
104	101, 103	3anbi23d 1435	. . . . 5 ⊢ (𝑚 = (^◡𝑓 “ ((1 / 2)[,]1)) → (((^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ ((^◡𝑓 “ (0[,](1 / 3))) ∩ 𝑚) = ∅) ↔ ((^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽) ∧ (^◡𝑓 “ ((1 / 2)[,]1)) ∈ (Clsd‘𝐽) ∧ ((^◡𝑓 “ (0[,](1 / 3))) ∩ (^◡𝑓 “ ((1 / 2)[,]1))) = ∅)))
105	100, 104	rspc2ev 3620	. . . 4 ⊢ (((^◡𝑓 “ (0[,](1 / 3))) ∈ ((nei‘𝐽)‘𝑆) ∧ (^◡𝑓 “ ((1 / 2)[,]1)) ∈ ((nei‘𝐽)‘𝑇) ∧ ((^◡𝑓 “ (0[,](1 / 3))) ∈ (Clsd‘𝐽) ∧ (^◡𝑓 “ ((1 / 2)[,]1)) ∈ (Clsd‘𝐽) ∧ ((^◡𝑓 “ (0[,](1 / 3))) ∩ (^◡𝑓 “ ((1 / 2)[,]1))) = ∅)) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
106	41, 70, 74, 78, 96, 105	syl113anc 1379	. . 3 ⊢ ((𝑓 ∈ (𝐽 Cn II) ∧ (𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1}))) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
107	106	rexlimiva 3136	. 2 ⊢ (∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (^◡𝑓 “ {0}) ∧ 𝑇 ⊆ (^◡𝑓 “ {1})) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
108	1, 107	syl 17	1 ⊢ (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∩ cin 3945 ⊆ wss 3946 ∅c0 4324 {csn 4632 ∪ cuni 4912 class class class wbr 5152 ^◡ccnv 5680 “ cima 5684 Fun wfun 6547 ‘cfv 6553 (class class class)co 7423 ℝcr 11153 0cc0 11154 1c1 11155 ℝ^*cxr 11293 < clt 11294 ≤ cle 11295 / cdiv 11917 2c2 12314 3c3 12315 (,]cioc 13374 [,)cico 13375 [,]cicc 13376 Topctop 22878 Clsdccld 23003 neicnei 23084 Cn ccn 23211 IIcii 24878

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fi 9450 df-sup 9481 df-inf 9482 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-q 12980 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-seq 14017 df-exp 14077 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-rest 17432 df-topgen 17453 df-ordt 17511 df-ps 18586 df-tsr 18587 df-psmet 21327 df-xmet 21328 df-met 21329 df-bl 21330 df-mopn 21331 df-top 22879 df-topon 22896 df-bases 22932 df-cld 23006 df-ntr 23007 df-nei 23085 df-cn 23214 df-ii 24880

This theorem is referenced by: (None)

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