| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → Ord 𝐵) | 
| 2 |  | ordelon 6408 | . . . . . . . . 9
⊢ ((Ord
𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) | 
| 3 | 2 | ancoms 458 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → 𝐴 ∈ On) | 
| 4 |  | ordeldifsucon 43272 | . . . . . . . 8
⊢ ((Ord
𝐵 ∧ 𝐴 ∈ On) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐))) | 
| 5 | 1, 3, 4 | syl2anc 584 | . . . . . . 7
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ 𝐵 ∧ 𝐴 ∈ 𝑐))) | 
| 6 | 5 | biancomd 463 | . . . . . 6
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))) | 
| 7 |  | ordelon 6408 | . . . . . . . . . 10
⊢ ((Ord
𝐵 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On) | 
| 8 | 7 | ad2ant2l 746 | . . . . . . . . 9
⊢ (((𝐴 ∈ 𝐵 ∧ Ord 𝐵) ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ On) | 
| 9 | 8 | ex 412 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ On)) | 
| 10 | 9 | pm4.71rd 562 | . . . . . . 7
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ (𝑐 ∈ On ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))) | 
| 11 |  | df-an 396 | . . . . . . 7
⊢ ((𝑐 ∈ On ∧ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))) | 
| 12 | 10, 11 | bitrdi 287 | . . . . . 6
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))) | 
| 13 | 6, 12 | bitr2d 280 | . . . . 5
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (¬ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)) ↔ 𝑐 ∈ (𝐵 ∖ suc 𝐴))) | 
| 14 | 13 | con1bid 355 | . . . 4
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ (𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))) | 
| 15 | 14 | albidv 1920 | . . 3
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵)))) | 
| 16 |  | eq0 4350 | . . 3
⊢ ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ¬ 𝑐 ∈ (𝐵 ∖ suc 𝐴)) | 
| 17 |  | df-ral 3062 | . . 3
⊢
(∀𝑐 ∈ On
¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) ↔ ∀𝑐(𝑐 ∈ On → ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))) | 
| 18 | 15, 16, 17 | 3bitr4g 314 | . 2
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ ↔ ∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵))) | 
| 19 |  | ordnexbtwnsuc 43280 | . 2
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴)) | 
| 20 | 18, 19 | sylbid 240 | 1
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴)) |