Proof of Theorem oaltom
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | om2 43417 | . . . . 5
⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o
2o)) | 
| 2 | 1 | ad2antlr 727 | . . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o
2o)) | 
| 3 |  | 2on 8520 | . . . . . . . 8
⊢
2o ∈ On | 
| 4 | 3 | a1i 11 | . . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o
∈ On) | 
| 5 |  | simpl 482 | . . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | 
| 6 |  | simpr 484 | . . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On) | 
| 7 | 4, 5, 6 | 3jca 1129 | . . . . . 6
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
(2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On)) | 
| 8 | 7 | adantr 480 | . . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧
𝐴 ∈ On ∧ 𝐵 ∈ On)) | 
| 9 |  | df-2o 8507 | . . . . . . 7
⊢
2o = suc 1o | 
| 10 | 9 | a1i 11 | . . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc
1o) | 
| 11 |  | simprl 771 | . . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴) | 
| 12 |  | eloni 6394 | . . . . . . . . . 10
⊢ (𝐴 ∈ On → Ord 𝐴) | 
| 13 | 12 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴) | 
| 14 | 13 | adantr 480 | . . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴) | 
| 15 | 11, 14 | jca 511 | . . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴)) | 
| 16 |  | ordelsuc 7840 | . . . . . . . 8
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o
⊆ 𝐴)) | 
| 17 | 16 | biimpd 229 | . . . . . . 7
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o
⊆ 𝐴)) | 
| 18 | 15, 11, 17 | sylc 65 | . . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴) | 
| 19 | 10, 18 | eqsstrd 4018 | . . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴) | 
| 20 |  | omwordi 8609 | . . . . 5
⊢
((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ⊆
𝐴 → (𝐵 ·o 2o) ⊆
(𝐵 ·o
𝐴))) | 
| 21 | 8, 19, 20 | sylc 65 | . . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 2o) ⊆
(𝐵 ·o
𝐴)) | 
| 22 | 2, 21 | eqsstrd 4018 | . . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴)) | 
| 23 | 6, 6 | jca 511 | . . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On)) | 
| 24 |  | simpr 484 | . . . 4
⊢
((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | 
| 25 |  | oaordi 8584 | . . . . 5
⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))) | 
| 26 | 25 | imp 406 | . . . 4
⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)) | 
| 27 | 23, 24, 26 | syl2an 596 | . . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)) | 
| 28 | 22, 27 | sseldd 3984 | . 2
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)) | 
| 29 | 28 | ex 412 | 1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
((1o ∈ 𝐴
∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))) |