Proof of Theorem ordsucelsuc
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . . 3
⊢ ((Ord
𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐵) | 
| 2 |  | ordelord 6405 | . . 3
⊢ ((Ord
𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | 
| 3 | 1, 2 | jca 511 | . 2
⊢ ((Ord
𝐵 ∧ 𝐴 ∈ 𝐵) → (Ord 𝐵 ∧ Ord 𝐴)) | 
| 4 |  | simpl 482 | . . 3
⊢ ((Ord
𝐵 ∧ suc 𝐴 ∈ suc 𝐵) → Ord 𝐵) | 
| 5 |  | ordsuc 7834 | . . . 4
⊢ (Ord
𝐵 ↔ Ord suc 𝐵) | 
| 6 |  | ordelord 6405 | . . . . 5
⊢ ((Ord suc
𝐵 ∧ suc 𝐴 ∈ suc 𝐵) → Ord suc 𝐴) | 
| 7 |  | ordsuc 7834 | . . . . 5
⊢ (Ord
𝐴 ↔ Ord suc 𝐴) | 
| 8 | 6, 7 | sylibr 234 | . . . 4
⊢ ((Ord suc
𝐵 ∧ suc 𝐴 ∈ suc 𝐵) → Ord 𝐴) | 
| 9 | 5, 8 | sylanb 581 | . . 3
⊢ ((Ord
𝐵 ∧ suc 𝐴 ∈ suc 𝐵) → Ord 𝐴) | 
| 10 | 4, 9 | jca 511 | . 2
⊢ ((Ord
𝐵 ∧ suc 𝐴 ∈ suc 𝐵) → (Ord 𝐵 ∧ Ord 𝐴)) | 
| 11 |  | ordsseleq 6412 | . . . . . . . 8
⊢ ((Ord suc
𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 12 | 7, 11 | sylanb 581 | . . . . . . 7
⊢ ((Ord
𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 13 | 12 | ancoms 458 | . . . . . 6
⊢ ((Ord
𝐵 ∧ Ord 𝐴) → (suc 𝐴 ⊆ 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 14 | 13 | adantl 481 | . . . . 5
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (suc 𝐴 ⊆ 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 15 |  | ordsucss 7839 | . . . . . . 7
⊢ (Ord
𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | 
| 16 | 15 | ad2antrl 728 | . . . . . 6
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | 
| 17 |  | sucssel 6478 | . . . . . . 7
⊢ (𝐴 ∈ V → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) | 
| 18 | 17 | adantr 480 | . . . . . 6
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) | 
| 19 | 16, 18 | impbid 212 | . . . . 5
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵)) | 
| 20 |  | sucexb 7825 | . . . . . . 7
⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | 
| 21 |  | elsucg 6451 | . . . . . . 7
⊢ (suc
𝐴 ∈ V → (suc
𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 22 | 20, 21 | sylbi 217 | . . . . . 6
⊢ (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 23 | 22 | adantr 480 | . . . . 5
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵))) | 
| 24 | 14, 19, 23 | 3bitr4d 311 | . . . 4
⊢ ((𝐴 ∈ V ∧ (Ord 𝐵 ∧ Ord 𝐴)) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) | 
| 25 | 24 | ex 412 | . . 3
⊢ (𝐴 ∈ V → ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))) | 
| 26 |  | elex 3500 | . . . . 5
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | 
| 27 |  | elex 3500 | . . . . . 6
⊢ (suc
𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V) | 
| 28 | 27, 20 | sylibr 234 | . . . . 5
⊢ (suc
𝐴 ∈ suc 𝐵 → 𝐴 ∈ V) | 
| 29 | 26, 28 | pm5.21ni 377 | . . . 4
⊢ (¬
𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) | 
| 30 | 29 | a1d 25 | . . 3
⊢ (¬
𝐴 ∈ V → ((Ord
𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵))) | 
| 31 | 25, 30 | pm2.61i 182 | . 2
⊢ ((Ord
𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) | 
| 32 | 3, 10, 31 | pm5.21nd 801 | 1
⊢ (Ord
𝐵 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ suc 𝐵)) |