Theorem dyadmax 24762

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 13681	. . . . 5 ⊢ < We (ℤ≥‘0)
2	1	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → < We (ℤ≥‘0))
3		nn0ex 12239	. . . . . 6 ⊢ ℕ0 ∈ V
4	3	rabex 5256	. . . . 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 4013	. . . . . 6 ⊢ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7		nn0uz 12620	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	sseqtri 3957	. . . . 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 24755	. . . . . . . . . . 11 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13		ffn 6600	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14		ovelrn 7448	. . . . . . . . . . 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 3234	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
17	15, 16	sylbb 218	. . . . . . . . 9 ⊢ (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
18	17	rgen 3074	. . . . . . . 8 ⊢ ∀𝑧 ∈ ran 𝐹∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19		ssralv 3987	. . . . . . . 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 4425	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
22	10, 20, 21	syl2anr 597	. . . . . 6 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23		rexcom 3234	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
24	22, 23	sylib 217	. . . . 5 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25		rabn0 4319	. . . . 5 ⊢ ({𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0 ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
26	24, 25	sylibr 233	. . . 4 ⊢ ((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27		wereu 5585	. . . 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 1371	. . 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 3362	. . 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 7283	. . . . . . 7 ⊢ (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
32	31	eqeq2d 2749	. . . . . 6 ⊢ (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33	32	2rexbidv 3229	. . . . 5 ⊢ (𝑛 = 𝑐 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
34	33	elrab 3624	. . . 4 ⊢ (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
37	36	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
38	35, 37	cbvrex2vw 3397	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
40	39	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41	40	2rexbidv 3229	. . . . . . . . 9 ⊢ (𝑛 = 𝑑 → (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
42	38, 41	bitrid 282	. . . . . . . 8 ⊢ (𝑛 = 𝑑 → (∃𝑧 ∈ 𝐴 ∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
43	42	ralrab 3630	. . . . . . 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 3208	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
45	44	ralbii 3092	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑑 ∈ ℕ0 ∀𝑤 ∈ 𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46		ralcom 3166	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ ℕ0 (∃𝑤 ∈ 𝐴 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤 ∈ 𝐴 ∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48		simplll 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
49	48	sselda 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹)
50		ovelrn 7448	. . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53		rexcom 3234	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0 ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54		r19.29 3184	. . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ℤ)) ∧ 𝑤 ∈ 𝐴) → 𝑎 ∈ ℤ)
59	58	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . 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 24761	. . . . . . . . . . . . . . . . . . . . . . . . 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 7284	. . . . . . . . . . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
70	69	sseq2d 3953	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71		eqeq2 2750	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
72	70, 71	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
73	72	imbi2d 341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
74	68, 73	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 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 649	. . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
86	85	sseq1d 3952	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
88	86, 87	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
89	88	ralbidv 3112	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑎𝐹𝑐) → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
90	84, 89	syl5ibrcom 246	. . . . . . . . . . 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 468	. . . . . . . . . 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 3213	. . . . . . . . 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 3203	. . . . . . . 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 413	. . . . . . 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 241	. . . . . 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 454	. . . 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 241	. . 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 3212	. 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ⟨cop 4567   class class class wbr 5074   We wwe 5543   × cxp 5587  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   / cdiv 11632  2c2 12028  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  [,]cicc 13082  ↑cexp 13782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628

This theorem is referenced by:  dyadmbllem  24763