Theorem dyadmax 25652

Description: Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)

Hypothesis

Ref	Expression
dyadmbl.1	⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)

Assertion

Ref	Expression
dyadmax	⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))

Distinct variable groups:   𝑥,𝑦   𝑧,𝑤,𝑥,𝑦,𝐴   𝑤,𝐹,𝑥,𝑦,𝑧

Proof of Theorem dyadmax

Dummy variables 𝑐 𝑑 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltweuz 14012	. . . . 5 ⊢ < We (ℤ≥‘0)
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → < We (ℤ≥‘0))
3		nn0ex 12559	. . . . . 6 ⊢ ℕ0 ∈ V
4	3	rabex 5357	. . . . 5 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6		ssrab2 4103	. . . . . 6 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7		nn0uz 12945	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	sseqtri 4045	. . . . 5 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ≥‘0)
9	8	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ≥‘0))
10		id 22	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → 𝐴 ≠ ∅)
11		dyadmbl.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
12	11	dyadf 25645	. . . . . . . . . . 11 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13		ffn 6747	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14		ovelrn 7626	. . . . . . . . . . 11 ⊢ (𝐹 Fn (ℤ × ℕ0) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛)))
15	12, 13, 14	mp2b 10	. . . . . . . . . 10 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛))
16		rexcom 3296	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
17	15, 16	sylbb 219	. . . . . . . . 9 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
18	17	rgen 3069	. . . . . . . 8 ⊢ ∀𝑧 ∈ ran 𝐹∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19		ssralv 4077	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (∀𝑧 ∈ ran 𝐹∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) → ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)))
20	18, 19	mpi 20	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
21		r19.2z 4518	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
22	10, 20, 21	syl2anr 596	. . . . . 6 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23		rexcom 3296	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
24	22, 23	sylib 218	. . . . 5 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25		rabn0 4412	. . . . 5 ⊢ ({𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
26	24, 25	sylibr 234	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27		wereu 5696	. . . 4 ⊢ (( < We (ℤ≥‘0) ∧ ({𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ≥‘0) ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
28	2, 5, 9, 26, 27	syl13anc 1372	. . 3 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29		reurex 3392	. . 3 ⊢ (∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
30	28, 29	syl 17	. 2 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
31		oveq2 7456	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
32	31	eqeq2d 2751	. . . . . 6 ⊢ (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33	32	2rexbidv 3228	. . . . 5 ⊢ (𝑛 = 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
34	33	elrab 3708	. . . 4 ⊢ (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35		eqeq1 2744	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
37	36	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
38	35, 37	cbvrex2vw 3248	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
40	39	eqeq2d 2751	. . . . . . . . . 10 ⊢ (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41	40	2rexbidv 3228	. . . . . . . . 9 ⊢ (𝑛 = 𝑑 → (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
42	38, 41	bitrid 283	. . . . . . . 8 ⊢ (𝑛 = 𝑑 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
43	42	ralrab 3715	. . . . . . 7 ⊢ (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44		r19.23v 3189	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
45	44	ralbii 3099	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46		ralcom 3295	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
47	45, 46	bitr3i 277	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48		simplll 774	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
49	48	sselda 4008	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
50		ovelrn 7626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Fn (ℤ × ℕ0) → (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑)))
51	12, 13, 50	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
52	49, 51	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53		rexcom 3296	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54		r19.29 3120	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
55	54	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
56	53, 55	sylbi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
57	52, 56	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
58		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑎 ∈ ℤ)
59	58	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑎 ∈ ℤ)
60		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑏 ∈ ℤ)
61		simp-5r 785	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑐 ∈ ℕ0)
62		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑑 ∈ ℕ0)
63		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ¬ 𝑑 < 𝑐)
64		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))
65	11, 59, 60, 61, 62, 63, 64	dyadmaxlem 25651	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏 ∧ 𝑐 = 𝑑))
66		oveq12 7457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝑏 ∧ 𝑐 = 𝑑) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
68	67	exp32 420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
69		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
70	69	sseq2d 4041	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71		eqeq2 2752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
72	70, 71	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
73	72	imbi2d 340	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
74	68, 73	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑑 ∈ ℕ0 ∧ 𝑏 ∈ ℤ)) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
75	74	anassrs 467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑑 ∈ ℕ0) ∧ 𝑏 ∈ ℤ) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
76	75	rexlimdva 3161	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑑 ∈ ℕ0) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
77	76	a2d 29	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑑 ∈ ℕ0) → ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
78	77	impd 410	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑑 ∈ ℕ0) → (((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
79	78	rexlimdva 3161	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → (∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
80	57, 79	syld 47	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
81	80	ralimdva 3173	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → (∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
82	47, 81	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
83	82	imp 406	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
84	83	an32s 651	. . . . . . . . . . . 12 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
85		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
86	85	sseq1d 4040	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
88	86, 87	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
89	88	ralbidv 3184	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎𝐹𝑐) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
90	84, 89	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
91	90	anassrs 467	. . . . . . . . . 10 ⊢ ((((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑎 ∈ ℤ) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
92	91	rexlimdva 3161	. . . . . . . . 9 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧 ∈ 𝐴) → (∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
93	92	reximdva 3174	. . . . . . . 8 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
94	93	ex 412	. . . . . . 7 ⊢ (((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
95	43, 94	biimtrid 242	. . . . . 6 ⊢ (((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
96	95	com23 86	. . . . 5 ⊢ (((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
97	96	expimpd 453	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ((𝑐 ∈ ℕ0 ∧ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
98	34, 97	biimtrid 242	. . 3 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
99	98	rexlimdv 3159	. 2 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
100	30, 99	mpd 15	1 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654   class class class wbr 5166   We wwe 5651   × cxp 5698  ran crn 5701   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   / cdiv 11947  2c2 12348  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  [,]cicc 13410  ↑cexp 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518

This theorem is referenced by:  dyadmbllem  25653