Theorem dyadmax 23659

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 12971	. . . . 5 ⊢ < We (ℤ≥‘0)
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → < We (ℤ≥‘0))
3		nn0ex 11547	. . . . . 6 ⊢ ℕ0 ∈ V
4	3	rabex 4975	. . . . 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 3849	. . . . . 6 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7		nn0uz 11925	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	sseqtri 3799	. . . . 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 23652	. . . . . . . . . . 11 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13		ffn 6225	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14		ovelrn 7010	. . . . . . . . . . 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 3246	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
17	15, 16	sylbb 210	. . . . . . . . 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 3828	. . . . . . . 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 4221	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
22	10, 20, 21	syl2anr 590	. . . . . 6 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23		rexcom 3246	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
24	22, 23	sylib 209	. . . . 5 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25		rabn0 4124	. . . . 5 ⊢ ({𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
26	24, 25	sylibr 225	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27		wereu 5275	. . . 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 1491	. . 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 3308	. . 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 6852	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
32	31	eqeq2d 2775	. . . . . 6 ⊢ (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33	32	2rexbidv 3204	. . . . 5 ⊢ (𝑛 = 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
34	33	elrab 3521	. . . 4 ⊢ (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35		eqeq1 2769	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36		oveq1 6851	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
37	36	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
38	35, 37	cbvrex2v 3328	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39		oveq2 6852	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
40	39	eqeq2d 2775	. . . . . . . . . 10 ⊢ (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41	40	2rexbidv 3204	. . . . . . . . 9 ⊢ (𝑛 = 𝑑 → (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
42	38, 41	syl5bb 274	. . . . . . . 8 ⊢ (𝑛 = 𝑑 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
43	42	ralrab 3527	. . . . . . 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 3170	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
45	44	ralbii 3127	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46		ralcom 3245	. . . . . . . . . . . . . . . 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 268	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48		simplll 791	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
49	48	sselda 3763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
50		ovelrn 7010	. . . . . . . . . . . . . . . . . . . 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 209	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53		rexcom 3246	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54		r19.29 3219	. . . . . . . . . . . . . . . . . . . 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 402	. . . . . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑎 ∈ ℤ)
59	58	ad2antrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . . . . . . . . . . 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 807	. . . . . . . . . . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . 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 23658	. . . . . . . . . . . . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . 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 6377	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
70	69	sseq2d 3795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71		eqeq2 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
72	70, 71	imbi12d 335	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
73	72	imbi2d 331	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
74	68, 73	syl5ibrcom 238	. . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . 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 398	. . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . . . 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	syl5bi 233	. . . . . . . . . . . . . 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 395	. . . . . . . . . . . . 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 642	. . . . . . . . . . . 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 6377	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
86	85	sseq1d 3794	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87		eqeq1 2769	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
88	86, 87	imbi12d 335	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
89	88	ralbidv 3133	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎𝐹𝑐) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
90	84, 89	syl5ibrcom 238	. . . . . . . . . . 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 459	. . . . . . . . . 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 3178	. . . . . . . . 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 3163	. . . . . . . 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 401	. . . . . . 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	syl5bi 233	. . . . . 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 445	. . . 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	syl5bi 233	. . 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 3177	. 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 197   ∧ wa 384   = wceq 1652   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  ∃!wreu 3057  {crab 3059  Vcvv 3350   ∩ cin 3733   ⊆ wss 3734  ∅c0 4081  ⟨cop 4342   class class class wbr 4811   We wwe 5237   × cxp 5277  ran crn 5280   Fn wfn 6065  ⟶wf 6066  ‘cfv 6070  (class class class)co 6844   ↦ cmpt2 6846  ℝcr 10190  0cc0 10191  1c1 10192   + caddc 10194   < clt 10330   ≤ cle 10331   / cdiv 10940  2c2 11329  ℕ₀cn0 11540  ℤcz 11626  ℤ_≥cuz 11889  [,]cicc 12383  ↑cexp 13070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fi 8526  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-sum 14705  df-rest 16352  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-top 20981  df-topon 20998  df-bases 21033  df-cmp 21473  df-ovol 23525

This theorem is referenced by:  dyadmbllem  23660