Theorem dyadmax 25718

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 13988	. . . . 5 ⊢ < We (ℤ≥‘0)
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → < We (ℤ≥‘0))
3		nn0ex 12501	. . . . . 6 ⊢ ℕ0 ∈ V
4	3	rabex 5300	. . . . 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 4036	. . . . . 6 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7		nn0uz 12891	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	sseqtri 3987	. . . . 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 23	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → 𝐴 ≠ ∅)
11		dyadmbl.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
12	11	dyadf 25711	. . . . . . . . . . 11 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13		ffn 6695	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14		ovelrn 7576	. . . . . . . . . . 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 3294	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
17	15, 16	sylbb 222	. . . . . . . . 9 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
18	17	rgen 3081	. . . . . . . 8 ⊢ ∀𝑧 ∈ ran 𝐹∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19		ssralv 4008	. . . . . . . 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 21	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
21		r19.2z 4456	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
22	10, 20, 21	syl2anr 608	. . . . . 6 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23		rexcom 3294	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
24	22, 23	sylib 221	. . . . 5 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25		rabn0 4346	. . . . 5 ⊢ ({𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
26	24, 25	sylibr 237	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27		wereu 5648	. . . 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 1395	. . 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 3374	. . 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 18	. 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 7408	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
32	31	eqeq2d 2776	. . . . . 6 ⊢ (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33	32	2rexbidv 3230	. . . . 5 ⊢ (𝑛 = 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
34	33	elrab 3653	. . . 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 7407	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
37	36	eqeq2d 2776	. . . . . . . . . 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 7408	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
40	39	eqeq2d 2776	. . . . . . . . . 10 ⊢ (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41	40	2rexbidv 3230	. . . . . . . . 9 ⊢ (𝑛 = 𝑑 → (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
42	38, 41	bitrid 286	. . . . . . . 8 ⊢ (𝑛 = 𝑑 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
43	42	ralrab 3660	. . . . . . 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 3192	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
45	44	ralbii 3111	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46		ralcom 3293	. . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48		simplll 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
49	48	sselda 3939	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
50		ovelrn 7576	. . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53		rexcom 3294	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54		r19.29 3128	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑎 ∈ ℤ)
59	58	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . . . . . . 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 25717	. . . . . . . . . . . . . . . . . . . . . . . . 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 7409	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑎 = 𝑏 ∧ 𝑐 = 𝑑) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
67	65, 66	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
70	69	sseq2d 3971	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71		eqeq2 2777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
72	70, 71	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
73	72	imbi2d 343	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
74	68, 73	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . . . . . . 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 30	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 3166	. . . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 664	. . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
86	85	sseq1d 3970	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87		eqeq1 2769	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
88	86, 87	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
89	88	ralbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎𝐹𝑐) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
90	84, 89	syl5ibrcom 250	. . . . . . . . . . 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 472	. . . . . . . . . 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 3166	. . . . . . . . 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 3178	. . . . . . . 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 417	. . . . . . 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 245	. . . . . 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 87	. . . . 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 458	. . . 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 245	. . 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 3164	. 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 16	1 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  ∃!wreu 3368  {crab 3417  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  ⟨cop 4591   class class class wbr 5105   We wwe 5604   × cxp 5650  ran crn 5653   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   < clt 11231   ≤ cle 11232   / cdiv 11859  2c2 12286  ℕ₀cn0 12495  ℤcz 12582  ℤ_≥cuz 12853  [,]cicc 13366  ↑cexp 14088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fi 9359  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ioo 13367  df-ico 13369  df-icc 13370  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-sum 15728  df-rest 17465  df-topgen 17486  df-psmet 21474  df-xmet 21475  df-met 21476  df-bl 21477  df-mopn 21478  df-top 23012  df-topon 23029  df-bases 23064  df-cmp 23505  df-ovol 25584

This theorem is referenced by:  dyadmbllem  25719