Step |
Hyp |
Ref |
Expression |
1 |
|
eleq2 |
⊢ ( 𝑥 = 𝑧 → ( 1 ∈ 𝑥 ↔ 1 ∈ 𝑧 ) ) |
2 |
|
eleq2 |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑦 + 1 ) ∈ 𝑥 ↔ ( 𝑦 + 1 ) ∈ 𝑧 ) ) |
3 |
2
|
raleqbi1dv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ↔ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) ) |
4 |
1 3
|
anbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ↔ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) ) ) |
5 |
|
dfnn2 |
⊢ ℕ = ∩ { 𝑧 ∣ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) } |
6 |
5
|
eqeq2i |
⊢ ( 𝑥 = ℕ ↔ 𝑥 = ∩ { 𝑧 ∣ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) } ) |
7 |
|
eleq2 |
⊢ ( 𝑥 = ℕ → ( 1 ∈ 𝑥 ↔ 1 ∈ ℕ ) ) |
8 |
|
eleq2 |
⊢ ( 𝑥 = ℕ → ( ( 𝑦 + 1 ) ∈ 𝑥 ↔ ( 𝑦 + 1 ) ∈ ℕ ) ) |
9 |
8
|
raleqbi1dv |
⊢ ( 𝑥 = ℕ → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ↔ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ ) ) |
10 |
7 9
|
anbi12d |
⊢ ( 𝑥 = ℕ → ( ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ↔ ( 1 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ ) ) ) |
11 |
6 10
|
sylbir |
⊢ ( 𝑥 = ∩ { 𝑧 ∣ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) } → ( ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ↔ ( 1 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ ) ) ) |
12 |
|
nnssre |
⊢ ℕ ⊆ ℝ |
13 |
5 12
|
eqsstrri |
⊢ ∩ { 𝑧 ∣ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) } ⊆ ℝ |
14 |
|
1nn |
⊢ 1 ∈ ℕ |
15 |
|
peano2nn |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 + 1 ) ∈ ℕ ) |
16 |
15
|
rgen |
⊢ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ |
17 |
14 16
|
pm3.2i |
⊢ ( 1 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ ) |
18 |
13 17
|
pm3.2i |
⊢ ( ∩ { 𝑧 ∣ ( 1 ∈ 𝑧 ∧ ∀ 𝑦 ∈ 𝑧 ( 𝑦 + 1 ) ∈ 𝑧 ) } ⊆ ℝ ∧ ( 1 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 + 1 ) ∈ ℕ ) ) |
19 |
4 11 18
|
intabs |
⊢ ∩ { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ) } = ∩ { 𝑥 ∣ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) } |
20 |
|
3anass |
⊢ ( ( 𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ↔ ( 𝑥 ⊆ ℝ ∧ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ) ) |
21 |
20
|
abbii |
⊢ { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) } = { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ) } |
22 |
21
|
inteqi |
⊢ ∩ { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) } = ∩ { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) ) } |
23 |
|
dfnn2 |
⊢ ℕ = ∩ { 𝑥 ∣ ( 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) } |
24 |
19 22 23
|
3eqtr4ri |
⊢ ℕ = ∩ { 𝑥 ∣ ( 𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 + 1 ) ∈ 𝑥 ) } |