Theorem ovoliunnfl 34949

Description: ovoliun 24106 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)

Hypothesis

Ref	Expression
ovoliunnfl.0	⊢ ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ))

Assertion

Ref	Expression
ovoliunnfl	⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Distinct variable group:   𝑓,𝑛,𝑚,𝑥,𝐴

Proof of Theorem ovoliunnfl

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4849	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4866	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	syl6eq 2872	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6674	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝐴) = (vol*‘∅))
5		ovol0 24094	. . . . . . 7 ⊢ (vol*‘∅) = 0
6	4, 5	syl6req 2873	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol*‘∪ 𝐴))
7	6	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
8		ovolge0 24082	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝐴))
9	8	ad2antll 727	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘∪ 𝐴))
10		reldom 8515	. . . . . . . . . . . 12 ⊢ Rel ≼
11	10	brrelex1i 5608	. . . . . . . . . . 11 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
12		0sdomg 8646	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
14	13	biimparc 482	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15		fodomr 8668	. . . . . . . . 9 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
16	14, 15	sylancom 590	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
17		unissb 4870	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ⊆ ℝ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ)
18	17	anbi1i 625	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
19		r19.26 3170	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
20	18, 19	bitr4i 280	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21		brdom2 8539	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22		nnenom 13349	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ≈ ω
23		sdomen2 8662	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25		isfinite 9115	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
26	24, 25	bitr4i 280	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
27	26	orbi1i 910	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
28	21, 27	bitri 277	. . . . . . . . . . . . 13 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29		ovolfi 24095	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
30	29	expcom 416	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31		ovolctb 24091	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
32	31	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
33	30, 32	jaod 855	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
34	28, 33	syl5bi 244	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
35	34	imdistani 571	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
36	35	ralimi 3160	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
37	20, 36	sylbi 219	. . . . . . . . 9 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
38	37	ancoms 461	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39		foima 6595	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑓 “ ℕ) = 𝐴)
40	39	raleqdv 3415	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41		fofn 6592	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓 Fn ℕ)
42		ssid 3989	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℕ
43		sseq1 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓‘𝑙) ⊆ ℝ))
44		fveqeq2 6679	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓‘𝑙)) = 0))
45	43, 44	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
46	45	ralima 7000	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
47	41, 42, 46	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
48	40, 47	bitr3d 283	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
49		fveq2 6670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (𝑓‘𝑙) = (𝑓‘𝑛))
50	49	sseq1d 3998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑛) ⊆ ℝ))
51		2fveq3 6675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑛)))
52	51	eqeq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑛)) = 0))
53	50, 52	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0)))
54	53	cbvralvw 3449	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0))
55		0re 10643	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
56		eleq1a 2908	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℝ → ((vol*‘(𝑓‘𝑛)) = 0 → (vol*‘(𝑓‘𝑛)) ∈ ℝ))
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((vol*‘(𝑓‘𝑛)) = 0 → (vol*‘(𝑓‘𝑛)) ∈ ℝ)
58	57	anim2i 618	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
59	58	ralimi 3160	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
60	54, 59	sylbi 219	. . . . . . . . . . . . . 14 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
61		ovoliunnfl.0	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ))
62	41, 60, 61	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ))
63		fofun 6591	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→𝐴 → Fun 𝑓)
64		funiunfv 7007	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝑓 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ (𝑓 “ ℕ))
65	63, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ (𝑓 “ ℕ))
66	39	unieqd 4852	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ (𝑓 “ ℕ) = ∪ 𝐴)
67	65, 66	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ 𝐴)
68	67	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
69	68	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
70		fveq2 6670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (𝑓‘𝑙) = (𝑓‘𝑚))
71	70	sseq1d 3998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ))
72		2fveq3 6675	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑚)))
73	72	eqeq1d 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑚)) = 0))
74	71, 73	anbi12d 632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑚 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0)))
75	74	rspccva 3622	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0))
76	75	simprd 498	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) = 0)
77	76	mpteq2dva 5161	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
78	77	seqeq3d 13378	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
79	78	rneqd 5808	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
80	79	supeq1d 8910	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ))
81		0cn 10633	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
82		ser1const 13427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
83	81, 82	mpan 688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84		nncn 11646	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
85	84	mul01d 10839	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
86	83, 85	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
87	86	mpteq2ia 5157	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88		fconstmpt 5614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89		seqeq3 13375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) → seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
90	88, 89	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0))
91		1z 12013	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
92		seqfn 13382	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1)
94		nnuz 12282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℤ≥‘1)
95	94	fneq2i 6451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
96		dffn5 6724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
97	95, 96	bitr3i 279	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (seq1( + , (ℕ × {0})) Fn (ℤ≥‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
98	93, 97	mpbi 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
99	90, 98	eqtr3i 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100		fconstmpt 5614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
101	87, 99, 100	3eqtr4i 2854	. . . . . . . . . . . . . . . . . . 19 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102	101	rneqi 5807	. . . . . . . . . . . . . . . . . 18 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103		1nn 11649	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
104		ne0i 4300	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
105		rnxp 6027	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ran (ℕ × {0}) = {0}
107	102, 106	eqtri 2844	. . . . . . . . . . . . . . . . 17 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108	107	supeq1i 8911	. . . . . . . . . . . . . . . 16 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109		xrltso 12535	. . . . . . . . . . . . . . . . 17 ⊢ < Or ℝ*
110		0xr 10688	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
111		supsn 8936	. . . . . . . . . . . . . . . . 17 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112	109, 110, 111	mp2an 690	. . . . . . . . . . . . . . . 16 ⊢ sup({0}, ℝ*, < ) = 0
113	108, 112	eqtri 2844	. . . . . . . . . . . . . . 15 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
114	80, 113	syl6eq 2872	. . . . . . . . . . . . . 14 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = 0)
115	114	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = 0)
116	62, 69, 115	3brtr3d 5097	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
117	116	ex 415	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (vol*‘∪ 𝐴) ≤ 0))
118	48, 117	sylbid 242	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
119	118	exlimiv 1931	. . . . . . . . 9 ⊢ (∃𝑓 𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
120	119	imp 409	. . . . . . . 8 ⊢ ((∃𝑓 𝑓:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
121	16, 38, 120	syl2an 597	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (vol*‘∪ 𝐴) ≤ 0)
122		ovolcl 24079	. . . . . . . . 9 ⊢ (∪ 𝐴 ⊆ ℝ → (vol*‘∪ 𝐴) ∈ ℝ*)
123		xrletri3 12548	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ (vol*‘∪ 𝐴) ∈ ℝ*) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
124	110, 122, 123	sylancr 589	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
125	124	ad2antll 727	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
126	9, 121, 125	mpbir2and 711	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
127	126	expl 460	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
128	7, 127	pm2.61ine 3100	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
129		renepnf 10689	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
130	55, 129	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
131		fveq2 6670	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = (vol*‘ℝ))
132		ovolre 24126	. . . . . . 7 ⊢ (vol*‘ℝ) = +∞
133	131, 132	syl6eq 2872	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = +∞)
134	130, 133	neeqtrrd 3090	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol*‘∪ 𝐴))
135	134	necon2i 3050	. . . 4 ⊢ (0 = (vol*‘∪ 𝐴) → ∪ 𝐴 ≠ ℝ)
136	128, 135	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → ∪ 𝐴 ≠ ℝ)
137	136	expr 459	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
138		eqimss 4023	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
139	138	necon3bi 3042	. 2 ⊢ (¬ ∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ)
140	137, 139	pm2.61d1 182	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ cuni 4838  ∪ ciun 4919   class class class wbr 5066   ↦ cmpt 5146   Or wor 5473   × cxp 5553  ran crn 5556   “ cima 5558  Fun wfun 6349   Fn wfn 6350  –onto→wfo 6353  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ≈ cen 8506   ≼ cdom 8507   ≺ csdm 8508  Fincfn 8509  supcsup 8904  ℂcc 10535  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542  +∞cpnf 10672  ℝ^*cxr 10674   < clt 10675   ≤ cle 10676  ℕcn 11638  ℤcz 11982  ℤ_≥cuz 12244  seqcseq 13370  vol*covol 24063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-z 11983  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-rest 16696  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24065

This theorem is referenced by:  ex-ovoliunnfl  34950