Theorem ovoliunnfl 36193

Description: ovoliun 24906 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 4901	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2787	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6851	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝐴) = (vol*‘∅))
5		ovol0 24894	. . . . . . 7 ⊢ (vol*‘∅) = 0
6	4, 5	eqtr2di 2788	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol*‘∪ 𝐴))
7	6	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
8		ovolge0 24882	. . . . . . . 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 8896	. . . . . . . . . . . 12 ⊢ Rel ≼
11	10	brrelex1i 5693	. . . . . . . . . . 11 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
12		0sdomg 9055	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
14	13	biimparc 480	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15		fodomr 9079	. . . . . . . . 9 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
16	14, 15	sylancom 588	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
17		unissb 4905	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ⊆ ℝ ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ)
18	17	anbi1i 624	. . . . . . . . . . 11 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
19		r19.26 3110	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
20	18, 19	bitr4i 277	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21		brdom2 8929	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22		nnenom 13895	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ≈ ω
23		sdomen2 9073	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25		isfinite 9597	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
26	24, 25	bitr4i 277	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
27	26	orbi1i 912	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
28	21, 27	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29		ovolfi 24895	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
30	29	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31		ovolctb 24891	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
32	31	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
33	30, 32	jaod 857	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
34	28, 33	biimtrid 241	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
35	34	imdistani 569	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
36	35	ralimi 3082	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
37	20, 36	sylbi 216	. . . . . . . . 9 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
38	37	ancoms 459	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39		foima 6766	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑓 “ ℕ) = 𝐴)
40	39	raleqdv 3311	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41		fofn 6763	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓 Fn ℕ)
42		ssid 3969	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℕ
43		sseq1 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓‘𝑙) ⊆ ℝ))
44		fveqeq2 6856	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓‘𝑙)) = 0))
45	43, 44	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑓‘𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
46	45	ralima 7193	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 280	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
49		fveq2 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (𝑓‘𝑙) = (𝑓‘𝑛))
50	49	sseq1d 3978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑛) ⊆ ℝ))
51		2fveq3 6852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑛)))
52	51	eqeq1d 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑛)) = 0))
53	50, 52	anbi12d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0)))
54	53	cbvralvw 3223	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0))
55		0re 11166	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
56		eleq1a 2827	. . . . . . . . . . . . . . . . . 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 617	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
59	58	ralimi 3082	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
60	54, 59	sylbi 216	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 6762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→𝐴 → Fun 𝑓)
64		funiunfv 7200	. . . . . . . . . . . . . . . . 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 4884	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ (𝑓 “ ℕ) = ∪ 𝐴)
67	65, 66	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ 𝐴)
68	67	fveq2d 6851	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
69	68	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
70		fveq2 6847	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (𝑓‘𝑙) = (𝑓‘𝑚))
71	70	sseq1d 3978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ))
72		2fveq3 6852	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑚)))
73	72	eqeq1d 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((vol*‘(𝑓‘𝑙)) = 0 ↔ (vol*‘(𝑓‘𝑚)) = 0))
74	71, 73	anbi12d 631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑚 → (((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0)))
75	74	rspccva 3581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0))
76	75	simprd 496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) = 0)
77	76	mpteq2dva 5210	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
78	77	seqeq3d 13924	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
79	78	rneqd 5898	. . . . . . . . . . . . . . . 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 9391	. . . . . . . . . . . . . . 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 11156	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
82		ser1const 13974	. . . . . . . . . . . . . . . . . . . . . . 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 12170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
85	84	mul01d 11363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
86	83, 85	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
87	86	mpteq2ia 5213	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88		fconstmpt 5699	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89		seqeq3 13921	. . . . . . . . . . . . . . . . . . . . . 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 12542	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
92		seqfn 13928	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1)
94		nnuz 12815	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℤ≥‘1)
95	94	fneq2i 6605	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
96		dffn5 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
97	95, 96	bitr3i 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (seq1( + , (ℕ × {0})) Fn (ℤ≥‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
98	93, 97	mpbi 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
99	90, 98	eqtr3i 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100		fconstmpt 5699	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
101	87, 99, 100	3eqtr4i 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102	101	rneqi 5897	. . . . . . . . . . . . . . . . . 18 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103		1nn 12173	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
104		ne0i 4299	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
105		rnxp 6127	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ran (ℕ × {0}) = {0}
107	102, 106	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108	107	supeq1i 9392	. . . . . . . . . . . . . . . 16 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109		xrltso 13070	. . . . . . . . . . . . . . . . 17 ⊢ < Or ℝ*
110		0xr 11211	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
111		supsn 9417	. . . . . . . . . . . . . . . . 17 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112	109, 110, 111	mp2an 690	. . . . . . . . . . . . . . . 16 ⊢ sup({0}, ℝ*, < ) = 0
113	108, 112	eqtri 2759	. . . . . . . . . . . . . . 15 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
114	80, 113	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = 0)
115	114	adantl 482	. . . . . . . . . . . . 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 5141	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
117	116	ex 413	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (vol*‘∪ 𝐴) ≤ 0))
118	48, 117	sylbid 239	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
119	118	exlimiv 1933	. . . . . . . . 9 ⊢ (∃𝑓 𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
120	119	imp 407	. . . . . . . 8 ⊢ ((∃𝑓 𝑓:ℕ–onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
121	16, 38, 120	syl2an 596	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (vol*‘∪ 𝐴) ≤ 0)
122		ovolcl 24879	. . . . . . . . 9 ⊢ (∪ 𝐴 ⊆ ℝ → (vol*‘∪ 𝐴) ∈ ℝ*)
123		xrletri3 13083	. . . . . . . . 9 ⊢ ((0 ∈ ℝ* ∧ (vol*‘∪ 𝐴) ∈ ℝ*) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
124	110, 122, 123	sylancr 587	. . . . . . . 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 458	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
128	7, 127	pm2.61ine 3024	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
129		renepnf 11212	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
130	55, 129	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
131		fveq2 6847	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = (vol*‘ℝ))
132		ovolre 24926	. . . . . . 7 ⊢ (vol*‘ℝ) = +∞
133	131, 132	eqtrdi 2787	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = +∞)
134	130, 133	neeqtrrd 3014	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol*‘∪ 𝐴))
135	134	necon2i 2974	. . . 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 457	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
138		eqimss 4005	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
139	138	necon3bi 2966	. 2 ⊢ (¬ ∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ)
140	137, 139	pm2.61d1 180	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  Vcvv 3446   ⊆ wss 3913  ∅c0 4287  {csn 4591  ∪ cuni 4870  ∪ ciun 4959   class class class wbr 5110   ↦ cmpt 5193   Or wor 5549   × cxp 5636  ran crn 5639   “ cima 5641  Fun wfun 6495   Fn wfn 6496  –onto→wfo 6499  ‘cfv 6501  (class class class)co 7362  ωcom 7807   ≈ cen 8887   ≼ cdom 8888   ≺ csdm 8889  Fincfn 8890  supcsup 9385  ℂcc 11058  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   · cmul 11065  +∞cpnf 11195  ℝ^*cxr 11197   < clt 11198   ≤ cle 11199  ℕcn 12162  ℤcz 12508  ℤ_≥cuz 12772  seqcseq 13916  vol*covol 24863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9356  df-sup 9387  df-inf 9388  df-oi 9455  df-dju 9846  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-q 12883  df-rp 12925  df-xneg 13042  df-xadd 13043  df-xmul 13044  df-ioo 13278  df-ico 13280  df-icc 13281  df-fz 13435  df-fzo 13578  df-seq 13917  df-exp 13978  df-hash 14241  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-clim 15382  df-sum 15583  df-rest 17318  df-topgen 17339  df-psmet 20825  df-xmet 20826  df-met 20827  df-bl 20828  df-mopn 20829  df-top 22280  df-topon 22297  df-bases 22333  df-cmp 22775  df-ovol 24865

This theorem is referenced by:  ex-ovoliunnfl  36194