Theorem ovoliunnfl 35828

Description: ovoliun 24678 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 4851	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4870	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2795	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6787	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝐴) = (vol*‘∅))
5		ovol0 24666	. . . . . . 7 ⊢ (vol*‘∅) = 0
6	4, 5	eqtr2di 2796	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol*‘∪ 𝐴))
7	6	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
8		ovolge0 24654	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝐴))
9	8	ad2antll 726	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘∪ 𝐴))
10		reldom 8748	. . . . . . . . . . . 12 ⊢ Rel ≼
11	10	brrelex1i 5644	. . . . . . . . . . 11 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
12		0sdomg 8900	. . . . . . . . . . 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 8924	. . . . . . . . 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 4874	. . . . . . . . . . . 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 3096	. . . . . . . . . . 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 8779	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22		nnenom 13709	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ≈ ω
23		sdomen2 8918	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25		isfinite 9419	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
26	24, 25	bitr4i 277	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
27	26	orbi1i 911	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
28	21, 27	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29		ovolfi 24667	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
30	29	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31		ovolctb 24663	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
32	31	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
33	30, 32	jaod 856	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
34	28, 33	syl5bi 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 3088	. . . . . . . . . 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 6702	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑓 “ ℕ) = 𝐴)
40	39	raleqdv 3349	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41		fofn 6699	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓 Fn ℕ)
42		ssid 3944	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℕ
43		sseq1 3947	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓‘𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓‘𝑙) ⊆ ℝ))
44		fveqeq2 6792	. . . . . . . . . . . . . . 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 7123	. . . . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (𝑓‘𝑙) = (𝑓‘𝑛))
50	49	sseq1d 3953	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑛) ⊆ ℝ))
51		2fveq3 6788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑛)))
52	51	eqeq1d 2741	. . . . . . . . . . . . . . . . 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 3384	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0))
55		0re 10986	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
56		eleq1a 2835	. . . . . . . . . . . . . . . . . 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 3088	. . . . . . . . . . . . . . 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 6698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→𝐴 → Fun 𝑓)
64		funiunfv 7130	. . . . . . . . . . . . . . . . 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 4854	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ (𝑓 “ ℕ) = ∪ 𝐴)
67	65, 66	eqtrd 2779	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ 𝐴)
68	67	fveq2d 6787	. . . . . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (𝑓‘𝑙) = (𝑓‘𝑚))
71	70	sseq1d 3953	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ))
72		2fveq3 6788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑚)))
73	72	eqeq1d 2741	. . . . . . . . . . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
78	77	seqeq3d 13738	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
79	78	rneqd 5850	. . . . . . . . . . . . . . . 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 9214	. . . . . . . . . . . . . . 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 10976	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
82		ser1const 13788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
83	81, 82	mpan 687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84		nncn 11990	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
85	84	mul01d 11183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
86	83, 85	eqtrd 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
87	86	mpteq2ia 5178	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88		fconstmpt 5650	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89		seqeq3 13735	. . . . . . . . . . . . . . . . . . . . . 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 12359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
92		seqfn 13742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1)
94		nnuz 12630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℤ≥‘1)
95	94	fneq2i 6540	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
96		dffn5 6837	. . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100		fconstmpt 5650	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
101	87, 99, 100	3eqtr4i 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102	101	rneqi 5849	. . . . . . . . . . . . . . . . . 18 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103		1nn 11993	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
104		ne0i 4269	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
105		rnxp 6078	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ran (ℕ × {0}) = {0}
107	102, 106	eqtri 2767	. . . . . . . . . . . . . . . . 17 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108	107	supeq1i 9215	. . . . . . . . . . . . . . . 16 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109		xrltso 12884	. . . . . . . . . . . . . . . . 17 ⊢ < Or ℝ*
110		0xr 11031	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
111		supsn 9240	. . . . . . . . . . . . . . . . 17 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112	109, 110, 111	mp2an 689	. . . . . . . . . . . . . . . 16 ⊢ sup({0}, ℝ*, < ) = 0
113	108, 112	eqtri 2767	. . . . . . . . . . . . . . 15 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
114	80, 113	eqtrdi 2795	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . 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 1934	. . . . . . . . 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 24651	. . . . . . . . 9 ⊢ (∪ 𝐴 ⊆ ℝ → (vol*‘∪ 𝐴) ∈ ℝ*)
123		xrletri3 12897	. . . . . . . . 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 726	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
126	9, 121, 125	mpbir2and 710	. . . . . 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 3029	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
129		renepnf 11032	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
130	55, 129	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
131		fveq2 6783	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = (vol*‘ℝ))
132		ovolre 24698	. . . . . . 7 ⊢ (vol*‘ℝ) = +∞
133	131, 132	eqtrdi 2795	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = +∞)
134	130, 133	neeqtrrd 3019	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol*‘∪ 𝐴))
135	134	necon2i 2979	. . . 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 3978	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
139	138	necon3bi 2971	. 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 844   = wceq 1539  ∃wex 1782   ∈ wcel 2107   ≠ wne 2944  ∀wral 3065  Vcvv 3433   ⊆ wss 3888  ∅c0 4257  {csn 4562  ∪ cuni 4840  ∪ ciun 4925   class class class wbr 5075   ↦ cmpt 5158   Or wor 5503   × cxp 5588  ran crn 5591   “ cima 5593  Fun wfun 6431   Fn wfn 6432  –onto→wfo 6435  ‘cfv 6437  (class class class)co 7284  ωcom 7721   ≈ cen 8739   ≼ cdom 8740   ≺ csdm 8741  Fincfn 8742  supcsup 9208  ℂcc 10878  ℝcr 10879  0cc0 10880  1c1 10881   + caddc 10883   · cmul 10885  +∞cpnf 11015  ℝ^*cxr 11017   < clt 11018   ≤ cle 11019  ℕcn 11982  ℤcz 12328  ℤ_≥cuz 12591  seqcseq 13730  vol*covol 24635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-inf2 9408  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-of 7542  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-2o 8307  df-er 8507  df-map 8626  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-fi 9179  df-sup 9210  df-inf 9211  df-oi 9278  df-dju 9668  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-n0 12243  df-z 12329  df-uz 12592  df-q 12698  df-rp 12740  df-xneg 12857  df-xadd 12858  df-xmul 12859  df-ioo 13092  df-ico 13094  df-icc 13095  df-fz 13249  df-fzo 13392  df-seq 13731  df-exp 13792  df-hash 14054  df-cj 14819  df-re 14820  df-im 14821  df-sqrt 14955  df-abs 14956  df-clim 15206  df-sum 15407  df-rest 17142  df-topgen 17163  df-psmet 20598  df-xmet 20599  df-met 20600  df-bl 20601  df-mopn 20602  df-top 22052  df-topon 22069  df-bases 22105  df-cmp 22547  df-ovol 24637

This theorem is referenced by:  ex-ovoliunnfl  35829