| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | omelon 9687 | . . 3
⊢ ω
∈ On | 
| 2 |  | peano1 7911 | . . . 4
⊢ ∅
∈ ω | 
| 3 | 2 | ne0ii 4343 | . . 3
⊢ ω
≠ ∅ | 
| 4 |  | omeu 8624 | . . 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 2576 | . . 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 7832 | . . . . . . . . . 10
⊢ (𝑎 ∈ On → suc 𝑎 ∈ On) | 
| 8 | 7 | adantr 480 | . . . . . . . . 9
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc
𝑎 ∈
On) | 
| 9 |  | simpr 484 | . . . . . . . . . 10
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ((ω
·o 𝑎)
+o 𝑏) = 𝐴) | 
| 10 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈
ω) | 
| 11 |  | simpl 482 | . . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) | 
| 12 |  | omcl 8575 | . . . . . . . . . . . . . . 15
⊢ ((ω
∈ On ∧ 𝑎 ∈
On) → (ω ·o 𝑎) ∈ On) | 
| 13 | 1, 11, 12 | sylancr 587 | . . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω
·o 𝑎)
∈ On) | 
| 14 |  | oaordi 8585 | . . . . . . . . . . . . . 14
⊢ ((ω
∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω
·o 𝑎)
+o 𝑏) ∈
((ω ·o 𝑎) +o ω))) | 
| 15 | 1, 13, 14 | sylancr 587 | . . . . . . . . . . . . 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 8565 | . . . . . . . . . . . . 13
⊢ ((ω
∈ On ∧ 𝑎 ∈
On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o
ω)) | 
| 18 | 1, 11, 17 | sylancr 587 | . . . . . . . . . . . 12
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω
·o suc 𝑎)
= ((ω ·o 𝑎) +o ω)) | 
| 19 | 16, 18 | eleqtrrd 2843 | . . . . . . . . . . 11
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) →
((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc
𝑎)) | 
| 20 | 19 | adantr 480 | . . . . . . . . . 10
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ((ω
·o 𝑎)
+o 𝑏) ∈
(ω ·o suc 𝑎)) | 
| 21 | 9, 20 | eqeltrrd 2841 | . . . . . . . . 9
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc
𝑎)) | 
| 22 |  | oveq2 7440 | . . . . . . . . . . 11
⊢ (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω
·o suc 𝑎)) | 
| 23 | 22 | eleq2d 2826 | . . . . . . . . . 10
⊢ (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc
𝑎))) | 
| 24 | 23 | rspcev 3621 | . . . . . . . . 9
⊢ ((suc
𝑎 ∈ On ∧ 𝐴 ∈ (ω
·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) | 
| 25 | 8, 21, 24 | syl2an2r 685 | . . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω
·o 𝑎)
+o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)) | 
| 26 | 25 | ex 412 | . . . . . . 7
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) →
(((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))) | 
| 27 | 26 | adantld 490 | . . . . . 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 1932 | . . 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)) |