Step | Hyp | Ref
| Expression |
1 | | omelon 9636 |
. . 3
⊢ ω
∈ On |
2 | | peano1 7873 |
. . . 4
⊢ ∅
∈ ω |
3 | 2 | ne0ii 4335 |
. . 3
⊢ ω
≠ ∅ |
4 | | omeu 8580 |
. . 3
⊢ ((ω
∈ On ∧ 𝐴 ∈ On
∧ ω ≠ ∅) → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴)) |
5 | 1, 3, 4 | mp3an13 1453 |
. 2
⊢ (𝐴 ∈ On → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴)) |
6 | | euex 2572 |
. . 3
⊢
(∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴)) |
7 | | onsuc 7793 |
. . . . . . . . . 10
⊢ (𝑎 ∈ On → suc 𝑎 ∈ On) |
8 | 7 | adantr 482 |
. . . . . . . . 9
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc
𝑎 ∈
On) |
9 | | simpr 486 |
. . . . . . . . . 10
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ((ω
·o 𝑎)
+o 𝑏) = 𝐴) |
10 | | simpr 486 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈
ω) |
11 | | simpl 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) |
12 | | omcl 8530 |
. . . . . . . . . . . . . . 15
⊢ ((ω
∈ On ∧ 𝑎 ∈
On) → (ω ·o 𝑎) ∈ On) |
13 | 1, 11, 12 | sylancr 588 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω
·o 𝑎)
∈ On) |
14 | | oaordi 8541 |
. . . . . . . . . . . . . 14
⊢ ((ω
∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω
·o 𝑎)
+o 𝑏) ∈
((ω ·o 𝑎) +o ω))) |
15 | 1, 13, 14 | sylancr 588 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (𝑏 ∈ ω → ((ω
·o 𝑎)
+o 𝑏) ∈
((ω ·o 𝑎) +o ω))) |
16 | 10, 15 | mpd 15 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) →
((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o
ω)) |
17 | | omsuc 8520 |
. . . . . . . . . . . . 13
⊢ ((ω
∈ On ∧ 𝑎 ∈
On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o
ω)) |
18 | 1, 11, 17 | sylancr 588 |
. . . . . . . . . . . 12
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω
·o suc 𝑎)
= ((ω ·o 𝑎) +o ω)) |
19 | 16, 18 | eleqtrrd 2837 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) →
((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc
𝑎)) |
20 | 19 | adantr 482 |
. . . . . . . . . 10
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ((ω
·o 𝑎)
+o 𝑏) ∈
(ω ·o suc 𝑎)) |
21 | 9, 20 | eqeltrrd 2835 |
. . . . . . . . 9
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc
𝑎)) |
22 | | oveq2 7411 |
. . . . . . . . . . 11
⊢ (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω
·o suc 𝑎)) |
23 | 22 | eleq2d 2820 |
. . . . . . . . . 10
⊢ (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc
𝑎))) |
24 | 23 | rspcev 3611 |
. . . . . . . . 9
⊢ ((suc
𝑎 ∈ On ∧ 𝐴 ∈ (ω
·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) |
25 | 8, 21, 24 | syl2an2r 684 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) |
26 | 25 | ex 414 |
. . . . . . 7
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) →
(((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) |
27 | 26 | adantld 492 |
. . . . . 6
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) |
28 | 27 | a1i 11 |
. . . . 5
⊢ (𝐴 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))) |
29 | 28 | rexlimdvv 3211 |
. . . 4
⊢ (𝐴 ∈ On → (∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) |
30 | 29 | exlimdv 1937 |
. . 3
⊢ (𝐴 ∈ On → (∃𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) |
31 | 6, 30 | syl5 34 |
. 2
⊢ (𝐴 ∈ On → (∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = 〈𝑎, 𝑏〉 ∧ ((ω ·o
𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) |
32 | 5, 31 | mpd 15 |
1
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) |