Theorem dyadmax 25925

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 14108	. . . . 5 ⊢ < We (ℤ≥‘0)
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → < We (ℤ≥‘0))
3		nn0ex 12616	. . . . . 6 ⊢ ℕ0 ∈ V
4	3	rabex 5299	. . . . 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 4005	. . . . . 6 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7		nn0uz 13007	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	sseqtri 3956	. . . . 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 25918	. . . . . . . . . . 11 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13		ffn 6710	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14		ovelrn 7598	. . . . . . . . . . 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 3291	. . . . . . . . . 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 3078	. . . . . . . 8 ⊢ ∀𝑧 ∈ ran 𝐹∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19		ssralv 3977	. . . . . . . 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 4454	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
22	10, 20, 21	syl2anr 609	. . . . . 6 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23		rexcom 3291	. . . . . 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 4338	. . . . 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 5646	. . . 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 1399	. . 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 3369	. . 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 7429	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
32	31	eqeq2d 2771	. . . . . 6 ⊢ (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33	32	2rexbidv 3227	. . . . 5 ⊢ (𝑛 = 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
34	33	elrab 3644	. . . 4 ⊢ (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
37	36	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
38	35, 37	cbvrex2vw 3245	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39		oveq2 7429	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
40	39	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41	40	2rexbidv 3227	. . . . . . . . 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 3651	. . . . . . 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 3108	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46		ralcom 3290	. . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
49	48	sselda 3908	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
50		ovelrn 7598	. . . . . . . . . . . . . . . . . . . 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 3291	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54		r19.29 3125	. . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑎 ∈ ℤ)
59	58	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . . . . 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 798	. . . . . . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . . . . . . . . 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 25924	. . . . . . . . . . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
70	69	sseq2d 3940	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 665	. . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
86	85	sseq1d 3939	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87		eqeq1 2764	. . . . . . . . . . . . . 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 3185	. . . . . . . . . . . 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 473	. . . . . . . . . 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 3163	. . . . . . . . 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 3175	. . . . . . . 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 418	. . . . . . 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 459	. . . 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 3161	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  ⟨cop 4834   class class class wbr 5102   We wwe 5602   × cxp 5648  ran crn 5651   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  ℝcr 11203  0cc0 11204  1c1 11205   + caddc 11207   < clt 11347   ≤ cle 11348   / cdiv 11977  2c2 12401  ℕ₀cn0 12610  ℤcz 12697  ℤ_≥cuz 12969  [,]cicc 13483  ↑cexp 14208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-map 8853  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-fi 9407  df-sup 9438  df-inf 9439  df-oi 9508  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-div 11978  df-nn 12340  df-2 12409  df-3 12410  df-n0 12611  df-z 12698  df-uz 12970  df-q 13080  df-rp 13125  df-xneg 13245  df-xadd 13246  df-xmul 13247  df-ioo 13484  df-ico 13486  df-icc 13487  df-fz 13644  df-fzo 13793  df-seq 14149  df-exp 14209  df-hash 14479  df-cj 15272  df-re 15273  df-im 15274  df-sqrt 15408  df-abs 15409  df-clim 15661  df-sum 15860  df-rest 17599  df-topgen 17620  df-psmet 21676  df-xmet 21677  df-met 21678  df-bl 21679  df-mopn 21680  df-top 23218  df-topon 23235  df-bases 23270  df-cmp 23711  df-ovol 25791

This theorem is used by:  dyadmbllem  25926