Theorem ovoliunnfl 38398

Description: ovoliun 25734 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 4881	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4899	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2813	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6886	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝐴) = (vol*‘∅))
5		ovol0 25722	. . . . . . 7 ⊢ (vol*‘∅) = 0
6	4, 5	eqtr2di 2814	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol*‘∪ 𝐴))
7	6	a1d 26	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
8		ovolge0 25710	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝐴))
9	8	ad2antll 742	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘∪ 𝐴))
10		reldom 8961	. . . . . . . . . . . 12 ⊢ Rel ≼
11	10	brrelex1i 5715	. . . . . . . . . . 11 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
12		0sdomg 9107	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
14	13	biimparc 485	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15		fodomr 9129	. . . . . . . . 9 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
16	14, 15	sylancom 600	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
17		unissb 4904	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ⊆ ℝ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ)
18	17	anbi1i 636	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
19		r19.26 3124	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
20	18, 19	bitr4i 281	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21		brdom2 8991	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22		nnenom 14046	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ≈ ω
23		sdomen2 9123	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25		isfinite 9634	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
26	24, 25	bitr4i 281	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
27	26	orbi1i 927	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
28	21, 27	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29		ovolfi 25723	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
30	29	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31		ovolctb 25719	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
32	31	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
33	30, 32	jaod 873	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
34	28, 33	biimtrid 245	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
35	34	imdistani 579	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
36	35	ralimi 3101	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
37	20, 36	sylbi 220	. . . . . . . . 9 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
38	37	ancoms 464	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39		foima 6798	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑓 “ ℕ) = 𝐴)
40	39	raleqdv 3321	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41		fofn 6795	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓 Fn ℕ)
42		ssid 3956	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℕ
43		sseq1 3959	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓‘𝑙) ⊆ ℝ))
44		fveqeq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓‘𝑙)) = 0))
45	43, 44	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
46	45	ralima 7239	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 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 284	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
49		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (𝑓‘𝑙) = (𝑓‘𝑛))
50	49	sseq1d 3965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑛) ⊆ ℝ))
51		2fveq3 6887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑛)))
52	51	eqeq1d 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑛)) = 0))
53	50, 52	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0)))
54	53	cbvralvw 3242	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0))
55		0re 11237	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
56		eleq1a 2857	. . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
59	58	ralimi 3101	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
60	54, 59	sylbi 220	. . . . . . . . . . . . . 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 608	. . . . . . . . . . . . 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 6794	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→𝐴 → Fun 𝑓)
64		funiunfv 7248	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝑓 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ (𝑓 “ ℕ))
65	63, 64	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ (𝑓 “ ℕ))
66	39	unieqd 4883	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ (𝑓 “ ℕ) = ∪ 𝐴)
67	65, 66	eqtrd 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ 𝐴)
68	67	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
69	68	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
70		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (𝑓‘𝑙) = (𝑓‘𝑚))
71	70	sseq1d 3965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ))
72		2fveq3 6887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑚)))
73	72	eqeq1d 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑚)) = 0))
74	71, 73	anbi12d 644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑚 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0)))
75	74	rspccva 3578	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0))
76	75	simprd 501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) = 0)
77	76	mpteq2dva 5202	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
78	77	seqeq3d 14075	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
79	78	rneqd 5926	. . . . . . . . . . . . . . . 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 9419	. . . . . . . . . . . . . . 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 11225	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
82		ser1const 14124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
83	81, 82	mpan 703	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84		nncn 12268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
85	84	mul01d 11436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
86	83, 85	eqtrd 2797	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
87	86	mpteq2ia 5204	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88		fconstmpt 5721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89		seqeq3 14072	. . . . . . . . . . . . . . . . . . . . . 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 12651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
92		seqfn 14079	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1)
94		nnuz 12929	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℤ≥‘1)
95	94	fneq2i 6634	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
96		dffn5 6940	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
97	95, 96	bitr3i 280	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (seq1( + , (ℕ × {0})) Fn (ℤ≥‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
98	93, 97	mpbi 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
99	90, 98	eqtr3i 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100		fconstmpt 5721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
101	87, 99, 100	3eqtr4i 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102	101	rneqi 5925	. . . . . . . . . . . . . . . . . 18 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103		1nn 12271	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
104		ne0i 4290	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
105		rnxp 6167	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ran (ℕ × {0}) = {0}
107	102, 106	eqtri 2785	. . . . . . . . . . . . . . . . 17 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108	107	supeq1i 9420	. . . . . . . . . . . . . . . 16 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109		xrltso 13194	. . . . . . . . . . . . . . . . 17 ⊢ < Or ℝ*
110		0xr 11283	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
111		supsn 9446	. . . . . . . . . . . . . . . . 17 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112	109, 110, 111	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ sup({0}, ℝ*, < ) = 0
113	108, 112	eqtri 2785	. . . . . . . . . . . . . . 15 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
114	80, 113	eqtrdi 2813	. . . . . . . . . . . . . 14 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = 0)
115	114	adantl 487	. . . . . . . . . . . . 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 5140	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
117	116	ex 418	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (vol*‘∪ 𝐴) ≤ 0))
118	48, 117	sylbid 243	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
119	118	exlimiv 1963	. . . . . . . . 9 ⊢ (∃𝑓 𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
120	119	imp 412	. . . . . . . 8 ⊢ ((∃𝑓 𝑓:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
121	16, 38, 120	syl2an 608	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (vol*‘∪ 𝐴) ≤ 0)
122		ovolcl 25707	. . . . . . . . 9 ⊢ (∪ 𝐴 ⊆ ℝ → (vol*‘∪ 𝐴) ∈ ℝ*)
123		xrletri3 13207	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ (vol*‘∪ 𝐴) ∈ ℝ*) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
124	110, 122, 123	sylancr 599	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
125	124	ad2antll 742	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
126	9, 121, 125	mpbir2and 726	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
127	126	expl 463	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
128	7, 127	pm2.61ine 3040	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
129		renepnf 11284	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
130	55, 129	mp1i 14	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
131		fveq2 6882	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = (vol*‘ℝ))
132		ovolre 25754	. . . . . . 7 ⊢ (vol*‘ℝ) = +∞
133	131, 132	eqtrdi 2813	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = +∞)
134	130, 133	neeqtrrd 3031	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol*‘∪ 𝐴))
135	134	necon2i 2991	. . . 4 ⊢ (0 = (vol*‘∪ 𝐴) → ∪ 𝐴 ≠ ℝ)
136	128, 135	syl 18	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → ∪ 𝐴 ≠ ℝ)
137	136	expr 462	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
138		eqimss 3992	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
139	138	necon3bi 2983	. 2 ⊢ (¬ ∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ)
140	137, 139	pm2.61d1 182	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  {csn 4587  ∪ cuni 4870  ∪ ciun 4954   class class class wbr 5107   ↦ cmpt 5190   Or wor 5566   × cxp 5657  ran crn 5660   “ cima 5662  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7416  ωcom 7865   ≈ cen 8952   ≼ cdom 8953   ≺ csdm 8954  Fincfn 8955  supcsup 9413  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130   · cmul 11132  +∞cpnf 11267  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271  ℕcn 12260  ℤcz 12618  ℤ_≥cuz 12890  seqcseq 14067  vol*covol 25691

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fi 9384  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-q 13001  df-rp 13045  df-xneg 13165  df-xadd 13166  df-xmul 13167  df-ioo 13404  df-ico 13406  df-icc 13407  df-fz 13564  df-fzo 13712  df-seq 14068  df-exp 14128  df-hash 14397  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-clim 15577  df-sum 15776  df-rest 17511  df-topgen 17532  df-psmet 21578  df-xmet 21579  df-met 21580  df-bl 21581  df-mopn 21582  df-top 23120  df-topon 23137  df-bases 23172  df-cmp 23613  df-ovol 25693

This theorem is used by:  ex-ovoliunnfl  38399