Theorem bcthlem5 25649

Description: Lemma for bcth 25650. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 14127, 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 25646) 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 25648). 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 25647) 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 25607	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3		metxmet 24653	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4	1, 2, 3	3syl 19	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
5		bcth.2	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘𝐷)
6	5	mopntop 24759	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	4, 6	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
8		bcthlem.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
9	8	frnd 6718	. . . . . . . 8 ⊢ (𝜑 → ran 𝑀 ⊆ (Clsd‘𝐽))
10		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
11	10	cldss2 23348	. . . . . . . 8 ⊢ (Clsd‘𝐽) ⊆ 𝒫 ∪ 𝐽
12	9, 11	sstrdi 3943	. . . . . . 7 ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 ∪ 𝐽)
13		sspwuni 5060	. . . . . . 7 ⊢ (ran 𝑀 ⊆ 𝒫 ∪ 𝐽 ↔ ∪ ran 𝑀 ⊆ ∪ 𝐽)
14	12, 13	sylib 221	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑀 ⊆ ∪ 𝐽)
15	10	ntropn 23367	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
16	7, 14, 15	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
17	4, 16	jca 521	. . . 4 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽))
18	5	mopni2 24812	. . . . 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 1136	. . . 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 592	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
21	5	mopnuni 24760	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
22	4, 21	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
23	10	topopn 23224	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
24	7, 23	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)
25	22, 24	eqeltrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
26		reex 11291	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		rpssre 13128	. . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℝ
28	26, 27	ssexi 5284	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
29		xpexg 7764	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
30	25, 28, 29	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 × ℝ+) ∈ V)
31	30	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (𝑋 × ℝ+) ∈ V)
32	10	ntrss3 23378	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
33	7, 14, 32	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
34	33, 22	sseqtrrd 3968	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
35	34	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
36		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
37	35, 36	sseldd 3932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ 𝑋)
38		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
39	37, 38	opelxpd 5690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
40		opabssxp 5743	. . . . . . . . . . . . 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 5295	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 261	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) → 𝑘 ∈ ℕ)
47		rspa 3252	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
48	45, 46, 47	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
49		ssdif0 4314	. . . . . . . . . . . . . . . . 17 ⊢ (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) ↔ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = ∅)
50		1st2nd2 8040	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
51	50	ad2antll 742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
52	51	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
53		df-ov 7423	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
54	52, 53	eqtr4di 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)))
55	4	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐷 ∈ (∞Met‘𝑋))
56		xp1st 8033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (1st ‘𝑧) ∈ 𝑋)
57	56	ad2antll 742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (1st ‘𝑧) ∈ 𝑋)
58		xp2nd 8034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (2nd ‘𝑧) ∈ ℝ+)
59	58	ad2antll 742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ+)
60		bln0 24734	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ+) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
61	55, 57, 59, 60	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
62	54, 61	eqnetrd 3023	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ≠ ∅)
63	7	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐽 ∈ Top)
64		ffvelcdm 7081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀:ℕ⟶(Clsd‘𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
65	8, 46, 64	syl2an 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
66	10	cldss 23347	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀‘𝑘) ∈ (Clsd‘𝐽) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
67	65, 66	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀‘𝑘) ⊆ ∪ 𝐽)
68	59	rpxrd 13165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ*)
69	5	blopn 24819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
70	55, 57, 68, 69	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
71	54, 70	eqeltrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ∈ 𝐽)
72	10	ssntr 23376	. . . . . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘))))
75		ssn0 4355	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)) ∧ ((ball‘𝐷)‘𝑧) ≠ ∅) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅)
76	75	expcom 419	. . . . . . . . . . . . . . . . . 18 ⊢ (((ball‘𝐷)‘𝑧) ≠ ∅ → (((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
77	62, 74, 76	sylsyld 62	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
78	49, 77	biimtrrid 246	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = ∅ → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅))
79	78	necon2d 2979	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((int‘𝐽)‘(𝑀‘𝑘)) = ∅ → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅))
80	48, 79	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅)
81		n0 4300	. . . . . . . . . . . . . . 15 ⊢ ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
82	4	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
83	10	difopn 23352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ (𝑀‘𝑘) ∈ (Clsd‘𝐽)) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
84	71, 65, 83	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
85	84	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
86		simp3 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
87		simp2l 1218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑘 ∈ ℕ)
88		nnrp 13132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
89	88	rpreccld 13174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ+)
90	87, 89	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (1 / 𝑘) ∈ ℝ+)
91	5	mopni3 24813	. . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . 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 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝜑)
94		elssuni 4899	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((ball‘𝐷)‘𝑧) ∈ 𝐽 → ((ball‘𝐷)‘𝑧) ⊆ ∪ 𝐽)
95	71, 94	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ ∪ 𝐽)
96	22	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑋 = ∪ 𝐽)
97	95, 96	sseqtrrd 3968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝑋)
98	97	ssdifssd 4094	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ⊆ 𝑋)
99	98	sseld 3930	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → 𝑥 ∈ 𝑋))
100	99	3impia 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ 𝑋)
101		simp2 1155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)))
102		rphalfcl 13149	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) ∈ ℝ+)
103		rphalflt 13151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) < 𝑛)
104		breq1 5106	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑟 = (𝑛 / 2) → (𝑟 < 𝑛 ↔ (𝑛 / 2) < 𝑛))
105	104	rspcev 3577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 / 2) ∈ ℝ+ ∧ (𝑛 / 2) < 𝑛) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
106	102, 103, 105	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℝ+ → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
107	106	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 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 3088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 ↔ ∃𝑟(𝑟 ∈ ℝ+ ∧ 𝑟 < 𝑛))
109		simpr3 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
110	109	rpred 13164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
111		simpr1 1213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
112	111	rpred 13164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ)
113		simplrl 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑘 ∈ ℕ)
114	113	nnrecred 12389	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (1 / 𝑘) ∈ ℝ)
115		simpr2 1214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 < 𝑛)
116		lttr 11386	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → ((𝑟 < 𝑛 ∧ 𝑛 < (1 / 𝑘)) → 𝑟 < (1 / 𝑘)))
117	116	expdimp 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) ∧ 𝑟 < 𝑛) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
118	110, 112, 114, 115, 117	syl31anc 1400	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
119	4	anim1i 627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
120	119	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
121		rpxr 13130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
122		rpxr 13130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ*)
123		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 < 𝑛 → 𝑟 < 𝑛)
124	121, 122, 123	3anim123i 1169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
125	124	3coml 1145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
126	5	blsscls 24826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 621	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑥 ∈ 𝑋)
132	131, 109	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+))
133	130, 132	jctild 535	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . 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 99	. . . . . . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . . . . . . 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 1950	. . . . . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . 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 1950	. . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 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 5528	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 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 4748	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 3206	. . . . . . . . . 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 8079	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
155	154	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
156		1z 12726	. . . . . . . . 9 ⊢ 1 ∈ ℤ
157		nnuz 13004	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
158	156, 157	axdc4uz 14127	. . . . . . . 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 1398	. . . . . . 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 1210	. . . . . . . . 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 18	. . . . . . . 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 18	. . . . . . . 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 1212	. . . . . . . 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 486	. . . . . . . 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 1213	. . . . . . . 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 1214	. . . . . . . 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 1215	. . . . . . . . 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 7443	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑔‘(𝑛 + 1)) = (𝑔‘(𝑘 + 1)))
169		id 23	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
170		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑔‘𝑛) = (𝑔‘𝑘))
171	169, 170	oveq12d 7438	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝐹(𝑔‘𝑛)) = (𝑘𝐹(𝑔‘𝑘)))
172	168, 171	eleq12d 2855	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
173	172	cbvralvw 3241	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
174	167, 173	sylib 221	. . . . . . . 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 25648	. . . . . . 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 1968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅)
177	10	ntrss2 23375	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
178	7, 14, 177	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
179		sstr2 3938	. . . . . . . . . 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 32	. . . . . . . . 9 ⊢ (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀) → (𝑛(ball‘𝐷)𝑚) ⊆ ∪ ran 𝑀))
181		ssdif0 4314	. . . . . . . . 9 ⊢ ((𝑛(ball‘𝐷)𝑚) ⊆ ∪ ran 𝑀 ↔ ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) = ∅)
182	180, 181	imbitrdi 254	. . . . . . . 8 ⊢ (𝜑 → ((𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) = ∅))
183	182	necon3ad 2969	. . . . . . 7 ⊢ (𝜑 → (((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀)))
184	183	3ad2ant1 1151	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
186	185	3expa 1136	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
187	186	nrexdv 3158	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ¬ ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
188	20, 187	pm2.65da 829	. 2 ⊢ (𝜑 → ¬ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
189	188	eq0rdv 4365	1 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5103  {copab 5167   × cxp 5649  ran crn 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  ℝcr 11199  1c1 11201   + caddc 11203  ℝ^*cxr 11342   < clt 11343   / cdiv 11973  ℕcn 12335  2c2 12397  ℝ⁺crp 13120  ∞Metcxmet 21663  Metcmet 21664  ballcbl 21665  MetOpencmopn 21668  Topctop 23211  Clsdccld 23334  intcnt 23335  clsccl 23336  CMetccmet 25575

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-dc 10524  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ico 13482  df-rest 17593  df-topgen 17614  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lm 23547  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-cfil 25576  df-cau 25577  df-cmet 25578

This theorem is used by:  bcth  25650