| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ssexg 5322 | . . . . . . . . 9
⊢ ((ω
⊆ 𝐵 ∧ 𝐵 ∈ On) → ω
∈ V) | 
| 2 | 1 | ex 412 | . . . . . . . 8
⊢ (ω
⊆ 𝐵 → (𝐵 ∈ On → ω ∈
V)) | 
| 3 |  | omelon2 7901 | . . . . . . . 8
⊢ (ω
∈ V → ω ∈ On) | 
| 4 | 2, 3 | syl6com 37 | . . . . . . 7
⊢ (𝐵 ∈ On → (ω
⊆ 𝐵 → ω
∈ On)) | 
| 5 | 4 | imp 406 | . . . . . 6
⊢ ((𝐵 ∈ On ∧ ω ⊆
𝐵) → ω ∈
On) | 
| 6 | 5 | adantll 714 | . . . . 5
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → ω
∈ On) | 
| 7 |  | simplr 768 | . . . . 5
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → 𝐵 ∈ On) | 
| 8 | 6, 7 | jca 511 | . . . 4
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → (ω
∈ On ∧ 𝐵 ∈
On)) | 
| 9 |  | oawordeu 8594 | . . . 4
⊢
(((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω
+o 𝑥) = 𝐵) | 
| 10 | 8, 9 | sylancom 588 | . . 3
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) →
∃!𝑥 ∈ On
(ω +o 𝑥) =
𝐵) | 
| 11 |  | reurex 3383 | . . 3
⊢
(∃!𝑥 ∈ On
(ω +o 𝑥) =
𝐵 → ∃𝑥 ∈ On (ω
+o 𝑥) = 𝐵) | 
| 12 | 10, 11 | syl 17 | . 2
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) →
∃𝑥 ∈ On (ω
+o 𝑥) = 𝐵) | 
| 13 |  | nnon 7894 | . . . . . . 7
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | 
| 14 | 13 | ad3antrrr 730 | . . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On) | 
| 15 | 6 | adantr 480 | . . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → ω
∈ On) | 
| 16 |  | simpr 484 | . . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On) | 
| 17 |  | oaass 8600 | . . . . . 6
⊢ ((𝐴 ∈ On ∧ ω ∈
On ∧ 𝑥 ∈ On)
→ ((𝐴 +o
ω) +o 𝑥) =
(𝐴 +o (ω
+o 𝑥))) | 
| 18 | 14, 15, 16, 17 | syl3anc 1372 | . . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω)
+o 𝑥) = (𝐴 +o (ω
+o 𝑥))) | 
| 19 |  | simpll 766 | . . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → 𝐴 ∈
ω) | 
| 20 |  | oaabslem 8686 | . . . . . . . 8
⊢ ((ω
∈ On ∧ 𝐴 ∈
ω) → (𝐴
+o ω) = ω) | 
| 21 | 6, 19, 20 | syl2anc 584 | . . . . . . 7
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → (𝐴 +o ω) =
ω) | 
| 22 | 21 | adantr 480 | . . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o ω) =
ω) | 
| 23 | 22 | oveq1d 7447 | . . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω)
+o 𝑥) = (ω
+o 𝑥)) | 
| 24 | 18, 23 | eqtr3d 2778 | . . . 4
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω
+o 𝑥)) =
(ω +o 𝑥)) | 
| 25 |  | oveq2 7440 | . . . . 5
⊢ ((ω
+o 𝑥) = 𝐵 → (𝐴 +o (ω +o 𝑥)) = (𝐴 +o 𝐵)) | 
| 26 |  | id 22 | . . . . 5
⊢ ((ω
+o 𝑥) = 𝐵 → (ω +o
𝑥) = 𝐵) | 
| 27 | 25, 26 | eqeq12d 2752 | . . . 4
⊢ ((ω
+o 𝑥) = 𝐵 → ((𝐴 +o (ω +o 𝑥)) = (ω +o
𝑥) ↔ (𝐴 +o 𝐵) = 𝐵)) | 
| 28 | 24, 27 | syl5ibcom 245 | . . 3
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω
+o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵)) | 
| 29 | 28 | rexlimdva 3154 | . 2
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) →
(∃𝑥 ∈ On
(ω +o 𝑥) =
𝐵 → (𝐴 +o 𝐵) = 𝐵)) | 
| 30 | 12, 29 | mpd 15 | 1
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω
⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵) |