| Step | Hyp | Ref
| Expression |
| 1 | | peano2re 11434 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈
ℝ) |
| 2 | | ltp1 12107 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ → 𝑛 < (𝑛 + 1)) |
| 3 | | breq2 5147 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑛 + 1) → (𝑛 < 𝑧 ↔ 𝑛 < (𝑛 + 1))) |
| 4 | 3 | rspcev 3622 |
. . . . . . . . . . . . 13
⊢ (((𝑛 + 1) ∈ ℝ ∧ 𝑛 < (𝑛 + 1)) → ∃𝑧 ∈ ℝ 𝑛 < 𝑧) |
| 5 | 1, 2, 4 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℝ →
∃𝑧 ∈ ℝ
𝑛 < 𝑧) |
| 6 | 5 | rgen 3063 |
. . . . . . . . . . 11
⊢
∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 |
| 7 | | ltnle 11340 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑛 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑛)) |
| 8 | 7 | rexbidva 3177 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℝ →
(∃𝑧 ∈ ℝ
𝑛 < 𝑧 ↔ ∃𝑧 ∈ ℝ ¬ 𝑧 ≤ 𝑛)) |
| 9 | | rexnal 3100 |
. . . . . . . . . . . . . 14
⊢
(∃𝑧 ∈
ℝ ¬ 𝑧 ≤ 𝑛 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
| 10 | 8, 9 | bitrdi 287 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ →
(∃𝑧 ∈ ℝ
𝑛 < 𝑧 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
| 11 | 10 | ralbiia 3091 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 ↔ ∀𝑛 ∈ ℝ ¬
∀𝑧 ∈ ℝ
𝑧 ≤ 𝑛) |
| 12 | | ralnex 3072 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
ℝ ¬ ∀𝑧
∈ ℝ 𝑧 ≤ 𝑛 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
| 13 | 11, 12 | bitri 275 |
. . . . . . . . . . 11
⊢
(∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
| 14 | 6, 13 | mpbi 230 |
. . . . . . . . . 10
⊢ ¬
∃𝑛 ∈ ℝ
∀𝑧 ∈ ℝ
𝑧 ≤ 𝑛 |
| 15 | | raleq 3323 |
. . . . . . . . . . 11
⊢ (𝐴 = ℝ → (∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛 ↔ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
| 16 | 15 | rexbidv 3179 |
. . . . . . . . . 10
⊢ (𝐴 = ℝ → (∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛 ↔ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
| 17 | 14, 16 | mtbiri 327 |
. . . . . . . . 9
⊢ (𝐴 = ℝ → ¬
∃𝑛 ∈ ℝ
∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
| 18 | | ssrab2 4080 |
. . . . . . . . . . . . . 14
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} |
| 19 | | ssrab2 4080 |
. . . . . . . . . . . . . . 15
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
| 20 | | zre 12617 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℝ) |
| 21 | | 2re 12340 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ |
| 22 | | reexpcl 14119 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℝ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
| 23 | 21, 22 | mpan 690 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℝ) |
| 24 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) |
| 25 | | 2cn 12341 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℂ |
| 26 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ≠
0 |
| 27 | | expne0i 14135 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0) |
| 28 | 25, 26, 27 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℤ →
(2↑𝑦) ≠
0) |
| 29 | 24, 28 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ≠
0) |
| 30 | 23, 29 | jca 511 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℝ ∧ (2↑𝑦)
≠ 0)) |
| 31 | | redivcl 11986 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℝ) |
| 32 | | peano2re 11434 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
| 33 | | redivcl 11986 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 + 1) ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
| 34 | 32, 33 | syl3an1 1164 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
| 35 | | opelxpi 5722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 / (2↑𝑦)) ∈ ℝ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 36 | 31, 34, 35 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 37 | 36 | 3expb 1121 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ℝ ∧
((2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 38 | 20, 30, 37 | syl2an 596 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 39 | 38 | rgen2 3199 |
. . . . . . . . . . . . . . . . 17
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ) |
| 40 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
| 41 | 40 | fmpo 8093 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ × ℝ)
↔ (𝑥 ∈ ℤ,
𝑦 ∈
ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ)) |
| 42 | 39, 41 | mpbi 230 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) |
| 43 | | frn 6743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ)) |
| 44 | 42, 43 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ) |
| 45 | 19, 44 | sstri 3993 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ ×
ℝ) |
| 46 | 18, 45 | sstri 3993 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ ×
ℝ) |
| 47 | | rnss 5950 |
. . . . . . . . . . . . . 14
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ) →
ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (ℝ ×
ℝ)) |
| 48 | | rnxpid 6193 |
. . . . . . . . . . . . . 14
⊢ ran
(ℝ × ℝ) = ℝ |
| 49 | 47, 48 | sseqtrdi 4024 |
. . . . . . . . . . . . 13
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ) →
ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ) |
| 50 | 46, 49 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ran
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ |
| 51 | | rnfi 9380 |
. . . . . . . . . . . 12
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) |
| 52 | | fimaxre2 12213 |
. . . . . . . . . . . 12
⊢ ((ran
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ ∧ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) |
| 53 | 50, 51, 52 | sylancr 587 |
. . . . . . . . . . 11
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) |
| 54 | 53 | adantl 481 |
. . . . . . . . . 10
⊢ ((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) |
| 55 | | eluni2 4911 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ↔ ∃𝑢 ∈ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ∈ 𝑢) |
| 56 | | iccf 13488 |
. . . . . . . . . . . . . . . . . . 19
⊢
[,]:(ℝ* × ℝ*)⟶𝒫
ℝ* |
| 57 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . 19
⊢
([,]:(ℝ* × ℝ*)⟶𝒫
ℝ* → [,] Fn (ℝ* ×
ℝ*)) |
| 58 | 56, 57 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ [,] Fn
(ℝ* × ℝ*) |
| 59 | | rexpssxrxp 11306 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
| 60 | 46, 59 | sstri 3993 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* ×
ℝ*) |
| 61 | | eleq2 2830 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = ([,]‘𝑣) → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ ([,]‘𝑣))) |
| 62 | 61 | rexima 7258 |
. . . . . . . . . . . . . . . . . 18
⊢ (([,] Fn
(ℝ* × ℝ*) ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* ×
ℝ*)) → (∃𝑢 ∈ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ∈ 𝑢 ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣))) |
| 63 | 58, 60, 62 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢
(∃𝑢 ∈
([,] “ {𝑎 ∈
{𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ∈ 𝑢 ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣)) |
| 64 | 55, 63 | bitri 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣)) |
| 65 | 46 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑣 ∈ (ℝ ×
ℝ)) |
| 66 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ (ℝ ×
ℝ) → 𝑣 =
〈(1st ‘𝑣), (2nd ‘𝑣)〉) |
| 67 | 66 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ([,]‘𝑣) = ([,]‘〈(1st
‘𝑣), (2nd
‘𝑣)〉)) |
| 68 | | df-ov 7434 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑣)[,](2nd ‘𝑣)) = ([,]‘〈(1st
‘𝑣), (2nd
‘𝑣)〉) |
| 69 | 67, 68 | eqtr4di 2795 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ([,]‘𝑣) = ((1st ‘𝑣)[,](2nd ‘𝑣))) |
| 70 | 69 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (𝑧 ∈
([,]‘𝑣) ↔ 𝑧 ∈ ((1st
‘𝑣)[,](2nd
‘𝑣)))) |
| 71 | 65, 70 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ([,]‘𝑣) ↔ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
| 72 | 71 | biimpd 229 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ([,]‘𝑣) → 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
| 73 | 72 | imdistani 568 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ([,]‘𝑣)) → (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
| 74 | | eliccxr 13475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ((1st
‘𝑣)[,](2nd
‘𝑣)) → 𝑧 ∈
ℝ*) |
| 75 | 74 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ∈ ℝ*) |
| 76 | | xp2nd 8047 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (2nd ‘𝑣) ∈ ℝ) |
| 77 | 76 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (2nd ‘𝑣) ∈
ℝ*) |
| 78 | 65, 77 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (2nd ‘𝑣) ∈
ℝ*) |
| 79 | 78 | ad2antrl 728 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → (2nd
‘𝑣) ∈
ℝ*) |
| 80 | | simpllr 776 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑛 ∈ ℝ) |
| 81 | 80 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑛 ∈ ℝ*) |
| 82 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (1st ‘𝑣) ∈ ℝ) |
| 83 | 82 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (1st ‘𝑣) ∈
ℝ*) |
| 84 | 83, 77 | jca 511 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ*)) |
| 85 | 65, 84 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((1st ‘𝑣) ∈ ℝ*
∧ (2nd ‘𝑣) ∈
ℝ*)) |
| 86 | | iccleub 13442 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ* ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
| 87 | 86 | 3expa 1119 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ*) ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
| 88 | 85, 87 | sylan 580 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
| 89 | 88 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ≤ (2nd ‘𝑣)) |
| 90 | | xpss 5701 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℝ
× ℝ) ⊆ (V × V) |
| 91 | 46, 90 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (V × V) |
| 92 | | df-rel 5692 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (Rel
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (V × V)) |
| 93 | 91, 92 | mpbir 231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ Rel
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} |
| 94 | | 2ndrn 8066 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Rel
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd ‘𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 95 | 93, 94 | mpan 690 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (2nd ‘𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 96 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 = (2nd ‘𝑣) → (𝑢 ≤ 𝑛 ↔ (2nd ‘𝑣) ≤ 𝑛)) |
| 97 | 96 | rspccva 3621 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((∀𝑢 ∈
ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛 ∧ (2nd ‘𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd ‘𝑣) ≤ 𝑛) |
| 98 | 95, 97 | sylan2 593 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑢 ∈
ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛 ∧ 𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd ‘𝑣) ≤ 𝑛) |
| 99 | 98 | ad2ant2lr 748 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → (2nd
‘𝑣) ≤ 𝑛) |
| 100 | 75, 79, 81, 89, 99 | xrletrd 13204 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ≤ 𝑛) |
| 101 | 73, 100 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ ((((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ([,]‘𝑣))) → 𝑧 ≤ 𝑛) |
| 102 | 101 | rexlimdvaa 3156 |
. . . . . . . . . . . . . . . 16
⊢ (((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) → (∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣) → 𝑧 ≤ 𝑛)) |
| 103 | 64, 102 | biimtrid 242 |
. . . . . . . . . . . . . . 15
⊢ (((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) → (𝑧 ∈ ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑧 ≤ 𝑛)) |
| 104 | 103 | ralrimiv 3145 |
. . . . . . . . . . . . . 14
⊢ (((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) → ∀𝑧 ∈ ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ≤ 𝑛) |
| 105 | | raleq 3323 |
. . . . . . . . . . . . . . 15
⊢ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 → (∀𝑧 ∈ ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ≤ 𝑛 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
| 106 | 105 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) → (∀𝑧 ∈ ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧 ≤ 𝑛 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
| 107 | 104, 106 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛) → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
| 108 | 107 | ex 412 |
. . . . . . . . . . . 12
⊢ ((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ 𝑛 ∈ ℝ) → (∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
| 109 | 108 | reximdva 3168 |
. . . . . . . . . . 11
⊢ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 → (∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
| 110 | 109 | adantr 480 |
. . . . . . . . . 10
⊢ ((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → (∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢 ≤ 𝑛 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
| 111 | 54, 110 | mpd 15 |
. . . . . . . . 9
⊢ ((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
| 112 | 17, 111 | nsyl 140 |
. . . . . . . 8
⊢ (𝐴 = ℝ → ¬ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) |
| 113 | 112 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
𝐴 = ℝ) → ¬
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) |
| 114 | | uniretop 24783 |
. . . . . . . . . . 11
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 115 | | retopconn 24851 |
. . . . . . . . . . . 12
⊢
(topGen‘ran (,)) ∈ Conn |
| 116 | 115 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → (topGen‘ran (,))
∈ Conn) |
| 117 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ∈ (topGen‘ran
(,))) |
| 118 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ≠ ∅) |
| 119 | | simprl 771 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴) |
| 120 | | ffun 6739 |
. . . . . . . . . . . . . . 15
⊢
([,]:(ℝ* × ℝ*)⟶𝒫
ℝ* → Fun [,]) |
| 121 | | funiunfv 7268 |
. . . . . . . . . . . . . . 15
⊢ (Fun [,]
→ ∪ 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) = ∪ ([,] “
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 122 | 56, 120, 121 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) = ∪ ([,] “
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 123 | | retop 24782 |
. . . . . . . . . . . . . . 15
⊢
(topGen‘ran (,)) ∈ Top |
| 124 | 46 | sseli 3979 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑧 ∈ (ℝ ×
ℝ)) |
| 125 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (ℝ ×
ℝ) → 𝑧 =
〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
| 126 | 125 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) = ([,]‘〈(1st
‘𝑧), (2nd
‘𝑧)〉)) |
| 127 | | df-ov 7434 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑧)[,](2nd ‘𝑧)) = ([,]‘〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
| 128 | 126, 127 | eqtr4di 2795 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) = ((1st ‘𝑧)[,](2nd ‘𝑧))) |
| 129 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → (1st ‘𝑧) ∈ ℝ) |
| 130 | | xp2nd 8047 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → (2nd ‘𝑧) ∈ ℝ) |
| 131 | | icccld 24787 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘𝑧) ∈ ℝ ∧ (2nd
‘𝑧) ∈ ℝ)
→ ((1st ‘𝑧)[,](2nd ‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 132 | 129, 130,
131 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ((1st ‘𝑧)[,](2nd ‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 133 | 128, 132 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 134 | 124, 133 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 135 | 134 | rgen 3063 |
. . . . . . . . . . . . . . 15
⊢
∀𝑧 ∈
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,))) |
| 136 | 114 | iuncld 23053 |
. . . . . . . . . . . . . . 15
⊢
(((topGen‘ran (,)) ∈ Top ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin ∧ ∀𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,))))
→ ∪ 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 137 | 123, 135,
136 | mp3an13 1454 |
. . . . . . . . . . . . . 14
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ∪ 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 138 | 122, 137 | eqeltrrid 2846 |
. . . . . . . . . . . . 13
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 139 | 138 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 140 | 119, 139 | eqeltrrd 2842 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ∈ (Clsd‘(topGen‘ran
(,)))) |
| 141 | 114, 116,
117, 118, 140 | connclo 23423 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
(∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 = ℝ) |
| 142 | 141 | ex 412 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ((∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → 𝐴 = ℝ)) |
| 143 | 142 | necon3ad 2953 |
. . . . . . . 8
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ (𝐴 ≠ ℝ
→ ¬ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))) |
| 144 | 143 | imp 406 |
. . . . . . 7
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
𝐴 ≠ ℝ) →
¬ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) |
| 145 | 113, 144 | pm2.61dane 3029 |
. . . . . 6
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ¬ (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) |
| 146 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦))) |
| 147 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1)) |
| 148 | 147 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦))) |
| 149 | 146, 148 | opeq12d 4881 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉) |
| 150 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣)) |
| 151 | 150 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣))) |
| 152 | 150 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣))) |
| 153 | 151, 152 | opeq12d 4881 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑣 → 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
| 154 | 149, 153 | cbvmpov 7528 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑢 ∈ ℤ, 𝑣 ∈ ℕ0 ↦
〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
| 155 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧)) |
| 156 | 155 | sseq1d 4015 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐))) |
| 157 | | equequ1 2024 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐)) |
| 158 | 156, 157 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
| 159 | 158 | ralbidv 3178 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
| 160 | 159 | cbvrabv 3447 |
. . . . . . . . . 10
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)} |
| 161 | 19 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐴 ∈ (topGen‘ran (,))
→ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 162 | 154, 160,
161 | dyadmbllem 25634 |
. . . . . . . . 9
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 163 | | opnmbllem0 37663 |
. . . . . . . . 9
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
| 164 | 162, 163 | eqtr3d 2779 |
. . . . . . . 8
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴) |
| 165 | 164 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴) |
| 166 | | nnenom 14021 |
. . . . . . . . 9
⊢ ℕ
≈ ω |
| 167 | | sdomentr 9151 |
. . . . . . . . 9
⊢ (({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ ∧ ℕ ≈
ω) → {𝑎 ∈
{𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω) |
| 168 | 166, 167 | mpan2 691 |
. . . . . . . 8
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω) |
| 169 | | isfinite 9692 |
. . . . . . . 8
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin ↔ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω) |
| 170 | 168, 169 | sylibr 234 |
. . . . . . 7
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) |
| 171 | 165, 170 | anim12i 613 |
. . . . . 6
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅) ∧
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ) → (∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) |
| 172 | 145, 171 | mtand 816 |
. . . . 5
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ¬ {𝑎 ∈
{𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ) |
| 173 | | qex 13003 |
. . . . . . . 8
⊢ ℚ
∈ V |
| 174 | 173, 173 | xpex 7773 |
. . . . . . 7
⊢ (ℚ
× ℚ) ∈ V |
| 175 | | zq 12996 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℚ) |
| 176 | | 2nn 12339 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℕ |
| 177 | | nnq 13004 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈
ℕ → 2 ∈ ℚ) |
| 178 | 176, 177 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℚ |
| 179 | | qexpcl 14118 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℚ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℚ) |
| 180 | 178, 179 | mpan 690 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℚ) |
| 181 | 180, 29 | jca 511 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℚ ∧ (2↑𝑦)
≠ 0)) |
| 182 | | qdivcl 13012 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℚ) |
| 183 | | 1z 12647 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℤ |
| 184 | | zq 12996 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
ℤ → 1 ∈ ℚ) |
| 185 | 183, 184 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℚ |
| 186 | | qaddcl 13007 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℚ ∧ 1 ∈
ℚ) → (𝑥 + 1)
∈ ℚ) |
| 187 | 185, 186 | mpan2 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℚ → (𝑥 + 1) ∈
ℚ) |
| 188 | | qdivcl 13012 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 + 1) ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℚ) |
| 189 | 187, 188 | syl3an1 1164 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℚ) |
| 190 | | opelxpi 5722 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 / (2↑𝑦)) ∈ ℚ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℚ) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
| 191 | 182, 189,
190 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
| 192 | 191 | 3expb 1121 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℚ ∧
((2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
| 193 | 175, 181,
192 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
| 194 | 193 | rgen2 3199 |
. . . . . . . . . . 11
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ) |
| 195 | 40 | fmpo 8093 |
. . . . . . . . . . 11
⊢
(∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ × ℚ)
↔ (𝑥 ∈ ℤ,
𝑦 ∈
ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℚ × ℚ)) |
| 196 | 194, 195 | mpbi 230 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℚ × ℚ) |
| 197 | | frn 6743 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℚ × ℚ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℚ
× ℚ)) |
| 198 | 196, 197 | ax-mp 5 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℚ
× ℚ) |
| 199 | 19, 198 | sstri 3993 |
. . . . . . . 8
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℚ ×
ℚ) |
| 200 | 18, 199 | sstri 3993 |
. . . . . . 7
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℚ ×
ℚ) |
| 201 | | ssdomg 9040 |
. . . . . . 7
⊢ ((ℚ
× ℚ) ∈ V → ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℚ × ℚ) →
{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ ×
ℚ))) |
| 202 | 174, 200,
201 | mp2 9 |
. . . . . 6
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ ×
ℚ) |
| 203 | | qnnen 16249 |
. . . . . . . 8
⊢ ℚ
≈ ℕ |
| 204 | | xpen 9180 |
. . . . . . . 8
⊢ ((ℚ
≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ)
≈ (ℕ × ℕ)) |
| 205 | 203, 203,
204 | mp2an 692 |
. . . . . . 7
⊢ (ℚ
× ℚ) ≈ (ℕ × ℕ) |
| 206 | | xpnnen 16247 |
. . . . . . 7
⊢ (ℕ
× ℕ) ≈ ℕ |
| 207 | 205, 206 | entri 9048 |
. . . . . 6
⊢ (ℚ
× ℚ) ≈ ℕ |
| 208 | | domentr 9053 |
. . . . . 6
⊢ (({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ × ℚ) ∧
(ℚ × ℚ) ≈ ℕ) → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ) |
| 209 | 202, 207,
208 | mp2an 692 |
. . . . 5
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ |
| 210 | 172, 209 | jctil 519 |
. . . 4
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ ∧ ¬ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ)) |
| 211 | | bren2 9023 |
. . . 4
⊢ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≈ ℕ ↔ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ ∧ ¬ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ)) |
| 212 | 210, 211 | sylibr 234 |
. . 3
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≈ ℕ) |
| 213 | 212 | ensymd 9045 |
. 2
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ℕ ≈ {𝑎
∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 214 | | bren 8995 |
. 2
⊢ (ℕ
≈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 215 | 213, 214 | sylib 218 |
1
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |