Theorem bcthlem5 25456

Description: Lemma for bcth 25457. The proof makes essential use of the Axiom of Dependent Choice axdc4uz 14020, 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 25453) 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 25455). 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 25454) 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 25414	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3		metxmet 24460	. . . . . 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 24566	. . . . . . 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 6715	. . . . . . . 8 ⊢ (𝜑 → ran 𝑀 ⊆ (Clsd‘𝐽))
10		eqid 2769	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
11	10	cldss2 23156	. . . . . . . 8 ⊢ (Clsd‘𝐽) ⊆ 𝒫 ∪ 𝐽
12	9, 11	sstrdi 3957	. . . . . . 7 ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 ∪ 𝐽)
13		sspwuni 5070	. . . . . . 7 ⊢ (ran 𝑀 ⊆ 𝒫 ∪ 𝐽 ↔ ∪ ran 𝑀 ⊆ ∪ 𝐽)
14	12, 13	sylib 221	. . . . . 6 ⊢ (𝜑 → ∪ ran 𝑀 ⊆ ∪ 𝐽)
15	10	ntropn 23175	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
16	7, 14, 15	syl2anc 595	. . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽)
17	4, 16	jca 520	. . . 4 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘∪ ran 𝑀) ∈ 𝐽))
18	5	mopni2 24619	. . . . 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 1134	. . . 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 591	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
21	5	mopnuni 24567	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
22	4, 21	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
23	10	topopn 23032	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
24	7, 23	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽)
25	22, 24	eqeltrd 2869	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
26		reex 11191	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		rpssre 13024	. . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℝ
28	26, 27	ssexi 5293	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
29		xpexg 7749	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ ℝ+ ∈ V) → (𝑋 × ℝ+) ∈ V)
30	25, 28, 29	sylancl 597	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 × ℝ+) ∈ V)
31	30	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → (𝑋 × ℝ+) ∈ V)
32	10	ntrss3 23186	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
33	7, 14, 32	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ 𝐽)
34	33, 22	sseqtrrd 3982	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
35	34	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ 𝑋)
36		simp2 1153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
37	35, 36	sseldd 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑛 ∈ 𝑋)
38		simp3 1154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
39	37, 38	opelxpd 5701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ⟨𝑛, 𝑚⟩ ∈ (𝑋 × ℝ+))
40		opabssxp 5754	. . . . . . . . . . . . 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 5304	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) → 𝑘 ∈ ℕ)
47		rspa 3260	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑘 ∈ ℕ ((int‘𝐽)‘(𝑀‘𝑘)) = ∅ ∧ 𝑘 ∈ ℕ) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
48	45, 46, 47	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((int‘𝐽)‘(𝑀‘𝑘)) = ∅)
49		ssdif0 4329	. . . . . . . . . . . . . . . . 17 ⊢ (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) ↔ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) = ∅)
50		1st2nd2 8025	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
51	50	ad2antll 741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
52	51	fveq2d 6886	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
53		df-ov 7414	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) = ((ball‘𝐷)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
54	52, 53	eqtr4di 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) = ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)))
55	4	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐷 ∈ (∞Met‘𝑋))
56		xp1st 8018	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (1st ‘𝑧) ∈ 𝑋)
57	56	ad2antll 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (1st ‘𝑧) ∈ 𝑋)
58		xp2nd 8019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝑋 × ℝ+) → (2nd ‘𝑧) ∈ ℝ+)
59	58	ad2antll 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ+)
60		bln0 24541	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ+) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
61	55, 57, 59, 60	syl3anc 1396	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ≠ ∅)
62	54, 61	eqnetrd 3031	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ≠ ∅)
63	7	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝐽 ∈ Top)
64		ffvelcdm 7077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀:ℕ⟶(Clsd‘𝐽) ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
65	8, 46, 64	syl2an 607	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑀‘𝑘) ∈ (Clsd‘𝐽))
66	10	cldss 23155	. . . . . . . . . . . . . . . . . . . 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 13061	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (2nd ‘𝑧) ∈ ℝ*)
69	5	blopn 24626	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ ℝ*) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
70	55, 57, 68, 69	syl3anc 1396	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((1st ‘𝑧)(ball‘𝐷)(2nd ‘𝑧)) ∈ 𝐽)
71	54, 70	eqeltrd 2869	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ∈ 𝐽)
72	10	ssntr 23184	. . . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ⊆ (𝑀‘𝑘) → ((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘))))
75		ssn0 4368	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ball‘𝐷)‘𝑧) ⊆ ((int‘𝐽)‘(𝑀‘𝑘)) ∧ ((ball‘𝐷)‘𝑧) ≠ ∅) → ((int‘𝐽)‘(𝑀‘𝑘)) ≠ ∅)
76	75	expcom 418	. . . . . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . . 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 4315	. . . . . . . . . . . . . . 15 ⊢ ((((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
82	4	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝐷 ∈ (∞Met‘𝑋))
83	10	difopn 23160	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ball‘𝐷)‘𝑧) ∈ 𝐽 ∧ (𝑀‘𝑘) ∈ (Clsd‘𝐽)) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
84	71, 65, 83	syl2anc 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
85	84	3adant3 1148	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ∈ 𝐽)
86		simp3 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)))
87		simp2l 1216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑘 ∈ ℕ)
88		nnrp 13028	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
89	88	rpreccld 13070	. . . . . . . . . . . . . . . . . . . 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 24620	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝜑)
94		elssuni 4908	. . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → 𝑋 = ∪ 𝐽)
97	95, 96	sseqtrrd 3982	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → ((ball‘𝐷)‘𝑧) ⊆ 𝑋)
98	97	ssdifssd 4109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) ⊆ 𝑋)
99	98	sseld 3944	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘)) → 𝑥 ∈ 𝑋))
100	99	3impia 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → 𝑥 ∈ 𝑋)
101		simp2 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)) ∧ 𝑥 ∈ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))) → (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+)))
102		rphalfcl 13045	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) ∈ ℝ+)
103		rphalflt 13047	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℝ+ → (𝑛 / 2) < 𝑛)
104		breq1 5116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑟 = (𝑛 / 2) → (𝑟 < 𝑛 ↔ (𝑛 / 2) < 𝑛))
105	104	rspcev 3590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 / 2) ∈ ℝ+ ∧ (𝑛 / 2) < 𝑛) → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
106	102, 103, 105	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℝ+ → ∃𝑟 ∈ ℝ+ 𝑟 < 𝑛)
107	106	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . 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 3096	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑟 ∈ ℝ+ 𝑟 < 𝑛 ↔ ∃𝑟(𝑟 ∈ ℝ+ ∧ 𝑟 < 𝑛))
109		simpr3 1213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
110	109	rpred 13060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ)
111		simpr1 1211	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
112	111	rpred 13060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑛 ∈ ℝ)
113		simplrl 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑘 ∈ ℕ)
114	113	nnrecred 12287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (1 / 𝑘) ∈ ℝ)
115		simpr2 1212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑟 < 𝑛)
116		lttr 11286	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → ((𝑟 < 𝑛 ∧ 𝑛 < (1 / 𝑘)) → 𝑟 < (1 / 𝑘)))
117	116	expdimp 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑟 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) ∧ 𝑟 < 𝑛) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
118	110, 112, 114, 115, 117	syl31anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑛 < (1 / 𝑘) → 𝑟 < (1 / 𝑘)))
119	4	anim1i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
120	119	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋))
121		rpxr 13026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
122		rpxr 13026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ*)
123		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑟 < 𝑛 → 𝑟 < 𝑛)
124	121, 122, 123	3anim123i 1167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑟 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
125	124	3coml 1143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+) → (𝑟 ∈ ℝ* ∧ 𝑛 ∈ ℝ* ∧ 𝑟 < 𝑛))
126	5	blsscls 24633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 607	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 620	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → 𝑥 ∈ 𝑋)
132	131, 109	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ (𝑋 × ℝ+))) ∧ (𝑛 ∈ ℝ+ ∧ 𝑟 < 𝑛 ∧ 𝑟 ∈ ℝ+)) → (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+))
133	130, 132	jctild 534	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . . 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 1944	. . . . . . . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . 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 1944	. . . . . . . . . . . . . . 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 5539	. . . . . . . . . . . . 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 4758	. . . . . . . . . . . 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 594	. . . . . . . . . . 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 3214	. . . . . . . . . 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 8065	. . . . . . . . . 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 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → 𝐹:(ℕ × (𝑋 × ℝ+))⟶(𝒫 (𝑋 × ℝ+) ∖ {∅}))
156		1z 12624	. . . . . . . . 9 ⊢ 1 ∈ ℤ
157		nnuz 12901	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
158	156, 157	axdc4uz 14020	. . . . . . . 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 1396	. . . . . . 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 1208	. . . . . . . . 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 1210	. . . . . . . 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 485	. . . . . . . 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 1211	. . . . . . . 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 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)))) → (𝑔‘1) = ⟨𝑛, 𝑚⟩)
167		simpr3 1213	. . . . . . . . 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 7434	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑔‘(𝑛 + 1)) = (𝑔‘(𝑘 + 1)))
169		id 23	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
170		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑔‘𝑛) = (𝑔‘𝑘))
171	169, 170	oveq12d 7429	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛𝐹(𝑔‘𝑛)) = (𝑘𝐹(𝑔‘𝑘)))
172	168, 171	eleq12d 2863	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘))))
173	172	cbvralvw 3249	. . . . . . . . 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 25455	. . . . . . 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 1962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀) ∧ 𝑚 ∈ ℝ+) → ((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅)
177	10	ntrss2 23183	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ∪ ran 𝑀 ⊆ ∪ 𝐽) → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
178	7, 14, 177	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) ⊆ ∪ ran 𝑀)
179		sstr2 3952	. . . . . . . . . 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 4329	. . . . . . . . 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 2977	. . . . . . 7 ⊢ (𝜑 → (((𝑛(ball‘𝐷)𝑚) ∖ ∪ ran 𝑀) ≠ ∅ → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀)))
184	183	3ad2ant1 1149	. . . . . 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 1134	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) ∧ 𝑚 ∈ ℝ+) → ¬ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
187	186	nrexdv 3166	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀)) → ¬ ∃𝑚 ∈ ℝ+ (𝑛(ball‘𝐷)𝑚) ⊆ ((int‘𝐽)‘∪ ran 𝑀))
188	20, 187	pm2.65da 828	. 2 ⊢ (𝜑 → ¬ 𝑛 ∈ ((int‘𝐽)‘∪ ran 𝑀))
189	188	eq0rdv 4378	1 ⊢ (𝜑 → ((int‘𝐽)‘∪ ran 𝑀) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ⟨cop 4600  ∪ cuni 4876   class class class wbr 5113  {copab 5177   × cxp 5660  ran crn 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  ℝcr 11099  1c1 11101   + caddc 11103  ℝ^*cxr 11242   < clt 11243   / cdiv 11871  ℕcn 12233  2c2 12295  ℝ⁺crp 13016  ∞Metcxmet 21476  Metcmet 21477  ballcbl 21478  MetOpencmopn 21481  Topctop 23019  Clsdccld 23142  intcnt 23143  clsccl 23144  CMetccmet 25382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-dc 10430  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9402  df-inf 9403  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ico 13378  df-rest 17475  df-topgen 17496  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-fbas 21488  df-fg 21489  df-top 23020  df-topon 23037  df-bases 23072  df-cld 23145  df-ntr 23146  df-cls 23147  df-nei 23224  df-lm 23355  df-fil 23972  df-fm 24064  df-flim 24065  df-flf 24066  df-cfil 25383  df-cau 25384  df-cmet 25385

This theorem is referenced by:  bcth  25457