Theorem ovoliunnfl 37656

Description: ovoliun 25406 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 4882	. . . . . . . . 9 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∪ ∅)
2		uni0 4899	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2780	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∪ 𝐴 = ∅)
4	3	fveq2d 6862	. . . . . . 7 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝐴) = (vol*‘∅))
5		ovol0 25394	. . . . . . 7 ⊢ (vol*‘∅) = 0
6	4, 5	eqtr2di 2781	. . . . . 6 ⊢ (𝐴 = ∅ → 0 = (vol*‘∪ 𝐴))
7	6	a1d 25	. . . . 5 ⊢ (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
8		ovolge0 25382	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝐴))
9	8	ad2antll 729	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘∪ 𝐴))
10		reldom 8924	. . . . . . . . . . . 12 ⊢ Rel ≼
11	10	brrelex1i 5694	. . . . . . . . . . 11 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
12		0sdomg 9070	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
14	13	biimparc 479	. . . . . . . . 9 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15		fodomr 9092	. . . . . . . . 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 4903	. . . . . . . . . . . 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 3091	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥 ∈ 𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ))
20	18, 19	bitr4i 278	. . . . . . . . . 10 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21		brdom2 8953	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22		nnenom 13945	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ≈ ω
23		sdomen2 9086	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25		isfinite 9605	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
26	24, 25	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
27	26	orbi1i 913	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
28	21, 27	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29		ovolfi 25395	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
30	29	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31		ovolctb 25391	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
32	31	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
33	30, 32	jaod 859	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
34	28, 33	biimtrid 242	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
35	34	imdistani 568	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
36	35	ralimi 3066	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
37	20, 36	sylbi 217	. . . . . . . . 9 ⊢ ((∪ 𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
38	37	ancoms 458	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ) → ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39		foima 6777	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑓 “ ℕ) = 𝐴)
40	39	raleqdv 3299	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41		fofn 6774	. . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 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 7211	. . . . . . . . . . . . 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 281	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)))
49		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (𝑓‘𝑙) = (𝑓‘𝑛))
50	49	sseq1d 3978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑛 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑛) ⊆ ℝ))
51		2fveq3 6863	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑛 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑛)))
52	51	eqeq1d 2731	. . . . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0))
55		0re 11176	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
56		eleq1a 2823	. . . . . . . . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ))
60	54, 59	sylbi 217	. . . . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→𝐴 → Fun 𝑓)
64		funiunfv 7222	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑚 ∈ ℕ (𝑓‘𝑚) = ∪ 𝐴)
68	67	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
69	68	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) = (vol*‘∪ 𝐴))
70		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (𝑓‘𝑙) = (𝑓‘𝑚))
71	70	sseq1d 3978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑚 → ((𝑓‘𝑙) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ))
72		2fveq3 6863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑚 → (vol*‘(𝑓‘𝑙)) = (vol*‘(𝑓‘𝑚)))
73	72	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . 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 3587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) = 0))
76	75	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) = 0)
77	76	mpteq2dva 5200	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ 0))
78	77	seqeq3d 13974	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
79	78	rneqd 5902	. . . . . . . . . . . . . . . 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 9397	. . . . . . . . . . . . . . 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 11166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
82		ser1const 14023	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
83	81, 82	mpan 690	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84		nncn 12194	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
85	84	mul01d 11373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
86	83, 85	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
87	86	mpteq2ia 5202	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88		fconstmpt 5700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89		seqeq3 13971	. . . . . . . . . . . . . . . . . . . . . 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 12563	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
92		seqfn 13978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
93	91, 92	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1)
94		nnuz 12836	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℤ≥‘1)
95	94	fneq2i 6616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ≥‘1))
96		dffn5 6919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
97	95, 96	bitr3i 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (seq1( + , (ℕ × {0})) Fn (ℤ≥‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
98	93, 97	mpbi 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
99	90, 98	eqtr3i 2754	. . . . . . . . . . . . . . . . . . . 20 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100		fconstmpt 5700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
101	87, 99, 100	3eqtr4i 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102	101	rneqi 5901	. . . . . . . . . . . . . . . . . 18 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103		1nn 12197	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
104		ne0i 4304	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
105		rnxp 6143	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ran (ℕ × {0}) = {0}
107	102, 106	eqtri 2752	. . . . . . . . . . . . . . . . 17 ⊢ ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108	107	supeq1i 9398	. . . . . . . . . . . . . . . 16 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109		xrltso 13101	. . . . . . . . . . . . . . . . 17 ⊢ < Or ℝ*
110		0xr 11221	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ*
111		supsn 9424	. . . . . . . . . . . . . . . . 17 ⊢ (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112	109, 110, 111	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ sup({0}, ℝ*, < ) = 0
113	108, 112	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
114	80, 113	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ) = 0)
115	114	adantl 481	. . . . . . . . . . . . 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 5138	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0)) → (vol*‘∪ 𝐴) ≤ 0)
117	116	ex 412	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑙 ∈ ℕ ((𝑓‘𝑙) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑙)) = 0) → (vol*‘∪ 𝐴) ≤ 0))
118	48, 117	sylbid 240	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
119	118	exlimiv 1930	. . . . . . . . 9 ⊢ (∃𝑓 𝑓:ℕ–onto→𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘∪ 𝐴) ≤ 0))
120	119	imp 406	. . . . . . . 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 25379	. . . . . . . . 9 ⊢ (∪ 𝐴 ⊆ ℝ → (vol*‘∪ 𝐴) ∈ ℝ*)
123		xrletri3 13114	. . . . . . . . 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 729	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → (0 = (vol*‘∪ 𝐴) ↔ (0 ≤ (vol*‘∪ 𝐴) ∧ (vol*‘∪ 𝐴) ≤ 0)))
126	9, 121, 125	mpbir2and 713	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
127	126	expl 457	. . . . 5 ⊢ (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴)))
128	7, 127	pm2.61ine 3008	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ (∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ ∧ ∪ 𝐴 ⊆ ℝ)) → 0 = (vol*‘∪ 𝐴))
129		renepnf 11222	. . . . . . 7 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
130	55, 129	mp1i 13	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → 0 ≠ +∞)
131		fveq2 6858	. . . . . . 7 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = (vol*‘ℝ))
132		ovolre 25426	. . . . . . 7 ⊢ (vol*‘ℝ) = +∞
133	131, 132	eqtrdi 2780	. . . . . 6 ⊢ (∪ 𝐴 = ℝ → (vol*‘∪ 𝐴) = +∞)
134	130, 133	neeqtrrd 2999	. . . . 5 ⊢ (∪ 𝐴 = ℝ → 0 ≠ (vol*‘∪ 𝐴))
135	134	necon2i 2959	. . . 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 456	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → (∪ 𝐴 ⊆ ℝ → ∪ 𝐴 ≠ ℝ))
138		eqimss 4005	. . 3 ⊢ (∪ 𝐴 = ℝ → ∪ 𝐴 ⊆ ℝ)
139	138	necon3bi 2951	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   class class class wbr 5107   ↦ cmpt 5188   Or wor 5545   × cxp 5636  ran crn 5639   “ cima 5641  Fun wfun 6505   Fn wfn 6506  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387  ωcom 7842   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917  Fincfn 8918  supcsup 9391  ℂcc 11066  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073  +∞cpnf 11205  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209  ℕcn 12186  ℤcz 12529  ℤ_≥cuz 12793  seqcseq 13966  vol*covol 25363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  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 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fi 9362  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-rest 17385  df-topgen 17406  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-mopn 21260  df-top 22781  df-topon 22798  df-bases 22833  df-cmp 23274  df-ovol 25365

This theorem is referenced by:  ex-ovoliunnfl  37657