Theorem bcthlem5 25299

Description: Lemma for bcth 25300. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 13919, which in the form used here accepts a "selection" function 𝐹 from each element of 𝐾 to a nonempty subset of 𝐾, and the result function 𝑔 maps 𝑔(𝑛 + 1) to an element of 𝐹(𝑛, 𝑔(𝑛)). The trick here is thus in the choice of 𝐹 and 𝐾: we let 𝐾 be the set of all tagged nonempty open sets (tagged here meaning that we have a point and an open set, in an ordered pair), and 𝐹(𝑘, ⟨𝑥, 𝑧⟩) gives the set of all balls of size less than 1 / 𝑘, tagged by their centers, whose closures fit within the given open set 𝑧 and miss 𝑀(𝑘).

Since 𝑀(𝑘) is closed, 𝑧 ∖ 𝑀(𝑘) is open and also nonempty, since 𝑧 is nonempty and 𝑀(𝑘) has empty interior. Then there is some ball contained in it, and hence our function 𝐹 is valid (it never maps to the empty set). Now starting at a point in the interior of ∪ ran 𝑀, DC gives us the function 𝑔 all whose elements are constrained by 𝐹 acting on the previous value. (This is all proven in this lemma.) Now 𝑔 is a sequence of tagged open balls, forming an inclusion chain (see bcthlem2 25296) and whose sizes tend to zero, since they are bounded above by 1 / 𝑘. Thus, the centers of these balls form a Cauchy sequence, and converge to a point 𝑥 (see bcthlem4 25298). Since the inclusion chain also ensures the closure of each ball is in the previous ball, the point 𝑥 must be in all these balls (see bcthlem3 25297) and hence misses each 𝑀(𝑘), contradicting the fact that 𝑥 is in the interior of ∪ ran 𝑀 (which was the starting point). (Contributed by Mario Carneiro, 6-Jan-2014.)

Hypotheses

Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem5.7	⊢ (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)

Assertion

Ref	Expression
bcthlem5	⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅)

Distinct variable groups:   𝑘,𝑟,𝑥,𝑧,𝐷   𝑘,𝐹,𝑟,𝑥,𝑧   𝑘,𝐽,𝑟,𝑥,𝑧   𝑘,𝑀,𝑟,𝑥,𝑧   𝜑,𝑘,𝑟,𝑥,𝑧   𝑘,𝑋,𝑟,𝑥,𝑧

Proof of Theorem bcthlem5

Dummy variables 𝑛 𝑔 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcthlem.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 25257	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3		metxmet 24293	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
5		bcth.2	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
6	5	mopntop 24399	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	4, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
8		bcthlem.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
9	8	frnd 6678	. . . . . . . 8 ⊢ (𝜑 → ran 𝑀 ⊆ (Clsd‘𝐽))
10		eqid 2737	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
11	10	cldss2 22989	. . . . . . . 8 ⊢ (Clsd‘𝐽) ⊆ 𝒫 ∪ 𝐽
12	9, 11	sstrdi 3948	. . . . . . 7 ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 ∪ 𝐽)
13		sspwuni 5057	. . . . . . 7 ⊢ (ran 𝑀 ⊆ 𝒫 ∪ 𝐽 ↔ ∪ ran 𝑀 ⊆ ∪ 𝐽)
14	12, 13	sylib 218	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑀 ⊆ ∪ 𝐽)
15	10	ntropn 23008	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
16	7, 14, 15	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
17	4, 16	jca 511	. . . 4 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽))
18	5	mopni2 24452	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
19	18	3expa 1119	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽) ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
20	17, 19	sylan 581	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
21	5	mopnuni 24400	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
22	4, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
23	10	topopn 22865	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
24	7, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)
25	22, 24	eqeltrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
26		reex 11129	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		rpssre 12925	. . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℝ
28	26, 27	ssexi 5269	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
29		xpexg 7705	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
30	25, 28, 29	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 × ℝ+) ∈ V)
31	30	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (𝑋 × ℝ+) ∈ V)
32	10	ntrss3 23019	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
33	7, 14, 32	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
34	33, 22	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
35	34	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
36		simp2 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
37	35, 36	sseldd 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ 𝑋)
38		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
39	37, 38	opelxpd 5671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
40		opabssxp 5724	. . . . . . . . . . . . 13 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ⊆ (𝑋 × ℝ+)
41		elpw2g 5280	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 × ℝ+) ∈ V → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ⊆ (𝑋 × ℝ+)))
42	30, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ⊆ (𝑋 × ℝ+)))
43	42	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ↔ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ⊆ (𝑋 × ℝ+)))
44	40, 43	mpbiri 258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ 𝒫 (𝑋 × ℝ+))
45		bcthlem5.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
46		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) → 𝑘 ∈ ℕ)
47		rspa 3227	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
48	45, 46, 47	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
49		ssdif0 4320	. . . . . . . . . . . . . . . . 17 ⊢ (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) ↔ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = ∅)
50		1st2nd2 7982	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
51	50	ad2antll 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
52	51	fveq2d 6846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
53		df-ov 7371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
54	52, 53	eqtr4di 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)))
55	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐷 ∈ (∞Met‘𝑋))
56		xp1st 7975	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (1st ‘𝑧) ∈ 𝑋)
57	56	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (1st ‘𝑧) ∈ 𝑋)
58		xp2nd 7976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (2nd ‘𝑧) ∈ ℝ+)
59	58	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ+)
60		bln0 24374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ+) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
61	55, 57, 59, 60	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
62	54, 61	eqnetrd 3000	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ≠ ∅)
63	7	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐽 ∈ Top)
64		ffvelcdm 7035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀:ℕ⟶(Clsd‘𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
65	8, 46, 64	syl2an 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
66	10	cldss 22988	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀‘𝑘) ∈ (Clsd‘𝐽) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
68	59	rpxrd 12962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ*)
69	5	blopn 24459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
70	55, 57, 68, 69	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
71	54, 70	eqeltrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ∈ 𝐽)
72	10	ssntr 23017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) ∧ (((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ ((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘))) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)))
73	72	expr 456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ (𝑀‘𝑘) ⊆ ∪ 𝐽) ∧ ((ball‘𝐷)‘𝑧) ∈ 𝐽) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘))))
74	63, 67, 71, 73	syl21anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘))))
75		ssn0 4358	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)) ∧ ((ball‘𝐷)‘𝑧) ≠ ∅) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅)
76	75	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((ball‘𝐷)‘𝑧) ≠ ∅ → (((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
77	62, 74, 76	sylsyld 61	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
78	49, 77	biimtrrid 243	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = ∅ → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
79	78	necon2d 2956	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅))
80	48, 79	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅)
81		n0 4307	. . . . . . . . . . . . . . 15 ⊢ ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
82	4	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
83	10	difopn 22993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ (𝑀‘𝑘) ∈ (Clsd‘𝐽)) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
84	71, 65, 83	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
85	84	3adant3 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
86		simp3 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
87		simp2l 1201	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑘 ∈ ℕ)
88		nnrp 12929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
89	88	rpreccld 12971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ+)
90	87, 89	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (1 / 𝑘) ∈ ℝ+)
91	5	mopni3 24453	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽 ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) ∧ (1 / 𝑘) ∈ ℝ+) → ∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))
92	82, 85, 86, 90, 91	syl31anc 1376	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))
93		simp1 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝜑)
94		elssuni 4896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((ball‘𝐷)‘𝑧) ∈ 𝐽 → ((ball‘𝐷)‘𝑧) ⊆ ∪ 𝐽)
95	71, 94	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ ∪ 𝐽)
96	22	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑋 = ∪ 𝐽)
97	95, 96	sseqtrrd 3973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝑋)
98	97	ssdifssd 4101	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ⊆ 𝑋)
99	98	sseld 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → 𝑥 ∈ 𝑋))
100	99	3impia 1118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ 𝑋)
101		simp2 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)))
102		rphalfcl 12946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) ∈ ℝ+)
103		rphalflt 12948	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) < 𝑛)
104		breq1 5103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑟 = (𝑛 / 2) → (𝑟 < 𝑛 ↔ (𝑛 / 2) < 𝑛))
105	104	rspcev 3578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 / 2) ∈ ℝ+ ∧ (𝑛 / 2) < 𝑛) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
106	102, 103, 105	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℝ+ → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
107	106	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
108		df-rex 3063	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 ↔ ∃𝑟(𝑟 ∈ ℝ+ ∧ 𝑟 < 𝑛))
109		simpr3 1198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
110	109	rpred 12961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
111		simpr1 1196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
112	111	rpred 12961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ)
113		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑘 ∈ ℕ)
114	113	nnrecred 12208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (1 / 𝑘) ∈ ℝ)
115		simpr2 1197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 < 𝑛)
116		lttr 11221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → ((𝑟 < 𝑛 ∧ 𝑛 < (1 / 𝑘)) → 𝑟 < (1 / 𝑘)))
117	116	expdimp 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) ∧ 𝑟 < 𝑛) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
118	110, 112, 114, 115, 117	syl31anc 1376	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
119	4	anim1i 616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
120	119	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
121		rpxr 12927	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
122		rpxr 12927	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ*)
123		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 < 𝑛 → 𝑟 < 𝑛)
124	121, 122, 123	3anim123i 1152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
125	124	3coml 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
126	5	blsscls 24466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
127	120, 125, 126	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛))
128		sstr2 3942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (𝑥(ball‘𝐷)𝑛) → ((𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))
129	127, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → ((𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))
130	118, 129	anim12d 610	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
131		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑥 ∈ 𝑋)
132	131, 109	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+))
133	130, 132	jctild 525	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
134	133	3exp2 1356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑛 ∈ ℝ+ → (𝑟 < 𝑛 → (𝑟 ∈ ℝ+ → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))))))
135	134	com35 98	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑛 ∈ ℝ+ → ((𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (𝑟 ∈ ℝ+ → (𝑟 < 𝑛 → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))))))
136	135	imp5d 439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) → ((𝑟 ∈ ℝ+ ∧ 𝑟 < 𝑛) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
137	136	eximdv 1919	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) → (∃𝑟(𝑟 ∈ ℝ+ ∧ 𝑟 < 𝑛) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
138	108, 137	biimtrid 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) → (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
139	107, 138	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ 𝑛 ∈ ℝ+) ∧ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
140	139	rexlimdva2 3141	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
141	93, 100, 101, 140	syl21anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (∃𝑛 ∈ ℝ+ (𝑛 < (1 / 𝑘) ∧ (𝑥(ball‘𝐷)𝑛) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
142	92, 141	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
143	142	3expia 1122	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → ∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
144	143	eximdv 1919	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → ∃𝑥∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
145	81, 144	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅ → ∃𝑥∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))))
146	80, 145	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ∃𝑥∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
147		opabn0 5509	. . . . . . . . . . . . 13 ⊢ ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ≠ ∅ ↔ ∃𝑥∃𝑟((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))))
148	146, 147	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ≠ ∅)
149		eldifsn 4744	. . . . . . . . . . . 12 ⊢ ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}) ↔ ({⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ 𝒫 (𝑋 × ℝ+) ∧ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ≠ ∅))
150	44, 148, 149	sylanbrc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}))
151	150	ralrimivva 3181	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ∀𝑧 ∈ (𝑋 × ℝ+){⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}))
152		bcthlem.5	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ+) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
153	152	fmpo 8022	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ ∀𝑧 ∈ (𝑋 × ℝ+){⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))} ∈ (𝒫 (𝑋 × ℝ+) ∖ {∅}) ↔ 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
154	151, 153	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
155	154	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
156		1z 12533	. . . . . . . . 9 ⊢ 1 ∈ ℤ
157		nnuz 12802	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
158	156, 157	axdc4uz 13919	. . . . . . . 8 ⊢ (((𝑋 × ℝ+) ∈ V ∧ ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+) ∧ 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅})) → ∃𝑔(𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
159	31, 39, 155, 158	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ∃𝑔(𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
160		simpl1 1193	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝜑)
161	160, 1	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝐷 ∈ (CMet‘𝑋))
162	160, 8	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝑀:ℕ⟶(Clsd‘𝐽))
163		simpl3 1195	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝑚 ∈ ℝ+)
164	37	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝑛 ∈ 𝑋)
165		simpr1 1196	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → 𝑔:ℕ⟶(𝑋 × ℝ+))
166		simpr2 1197	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → (𝑔‘1) = ⟨𝑛, 𝑚⟩)
167		simpr3 1198	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
168		fvoveq1 7391	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑔‘(𝑛 + 1)) = (𝑔‘(𝑘 + 1)))
169		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
170		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑔‘𝑛) = (𝑔‘𝑘))
171	169, 170	oveq12d 7386	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝐹(𝑔‘𝑛)) = (𝑘𝐹(𝑔‘𝑘)))
172	168, 171	eleq12d 2831	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
173	172	cbvralvw 3216	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
174	167, 173	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
175	5, 161, 152, 162, 163, 164, 165, 166, 174	bcthlem4 25298	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) ∧ (𝑔:ℕ⟶(𝑋 × ℝ+) ∧ (𝑔‘1) = ⟨𝑛, 𝑚⟩ ∧ ∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅)
176	159, 175	exlimddv 1937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅)
177	10	ntrss2 23016	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
178	7, 14, 177	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
179		sstr2 3942	. . . . . . . . . 10 ⊢ ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀) → (((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀 → (𝑛(ball‘𝐷)𝑚) ⊆ ∪ ran 𝑀))
180	178, 179	syl5com 31	. . . . . . . . 9 ⊢ (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀) → (𝑛(ball‘𝐷)𝑚) ⊆ ∪ ran 𝑀))
181		ssdif0 4320	. . . . . . . . 9 ⊢ ((𝑛(ball‘𝐷)𝑚) ⊆ ∪ ran 𝑀 ↔ ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) = ∅)
182	180, 181	imbitrdi 251	. . . . . . . 8 ⊢ (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) = ∅))
183	182	necon3ad 2946	. . . . . . 7 ⊢ (𝜑 → (((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀)))
184	183	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀)))
185	176, 184	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
186	185	3expa 1119	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
187	186	nrexdv 3133	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ¬ ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
188	20, 187	pm2.65da 817	. 2 ⊢ (𝜑 → ¬ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
189	188	eq0rdv 4361	1 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100  {copab 5162   × cxp 5630  ran crn 5633  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  ℝcr 11037  1c1 11039   + caddc 11041  ℝ^*cxr 11177   < clt 11178   / cdiv 11806  ℕcn 12157  2c2 12212  ℝ⁺crp 12917  ∞Metcxmet 21309  Metcmet 21310  ballcbl 21311  MetOpencmopn 21314  Topctop 22852  Clsdccld 22975  intcnt 22976  clsccl 22977  CMetccmet 25225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-dc 10368  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ico 13279  df-rest 17354  df-topgen 17375  df-psmet 21316  df-xmet 21317  df-met 21318  df-bl 21319  df-mopn 21320  df-fbas 21321  df-fg 21322  df-top 22853  df-topon 22870  df-bases 22905  df-cld 22978  df-ntr 22979  df-cls 22980  df-nei 23057  df-lm 23188  df-fil 23805  df-fm 23897  df-flim 23898  df-flf 23899  df-cfil 25226  df-cau 25227  df-cmet 25228

This theorem is referenced by:  bcth  25300