| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | onelon 6408 | . . . . 5
⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) | 
| 2 | 1 | adantll 714 | . . . 4
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On) | 
| 3 |  | eloni 6393 | . . . . . . . . 9
⊢ (𝐵 ∈ On → Ord 𝐵) | 
| 4 |  | ordsucss 7839 | . . . . . . . . 9
⊢ (Ord
𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | 
| 5 | 3, 4 | syl 17 | . . . . . . . 8
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | 
| 6 | 5 | ad2antlr 727 | . . . . . . 7
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) | 
| 7 |  | onsucb 7838 | . . . . . . . . . 10
⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | 
| 8 |  | oveq2 7440 | . . . . . . . . . . . . . 14
⊢ (𝑥 = suc 𝐴 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝐴)) | 
| 9 | 8 | sseq2d 4015 | . . . . . . . . . . . . 13
⊢ (𝑥 = suc 𝐴 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴))) | 
| 10 | 9 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑥 = suc 𝐴 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴)))) | 
| 11 |  | oveq2 7440 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o 𝑦)) | 
| 12 | 11 | sseq2d 4015 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))) | 
| 13 | 12 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)))) | 
| 14 |  | oveq2 7440 | . . . . . . . . . . . . . 14
⊢ (𝑥 = suc 𝑦 → (𝐶 +o 𝑥) = (𝐶 +o suc 𝑦)) | 
| 15 | 14 | sseq2d 4015 | . . . . . . . . . . . . 13
⊢ (𝑥 = suc 𝑦 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))) | 
| 16 | 15 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑥 = suc 𝑦 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))) | 
| 17 |  | oveq2 7440 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝐵 → (𝐶 +o 𝑥) = (𝐶 +o 𝐵)) | 
| 18 | 17 | sseq2d 4015 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝐵 → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥) ↔ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))) | 
| 19 | 18 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝐵 → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) ↔ (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵)))) | 
| 20 |  | ssid 4005 | . . . . . . . . . . . . 13
⊢ (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴) | 
| 21 | 20 | 2a1i 12 | . . . . . . . . . . . 12
⊢ (suc
𝐴 ∈ On → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝐴))) | 
| 22 |  | sssucid 6463 | . . . . . . . . . . . . . . . . 17
⊢ (𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦) | 
| 23 |  | sstr2 3989 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → ((𝐶 +o 𝑦) ⊆ suc (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))) | 
| 24 | 22, 23 | mpi 20 | . . . . . . . . . . . . . . . 16
⊢ ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦)) | 
| 25 |  | oasuc 8563 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐶 ∈ On ∧ 𝑦 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦)) | 
| 26 | 25 | ancoms 458 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝑦) = suc (𝐶 +o 𝑦)) | 
| 27 | 26 | sseq2d 4015 | . . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦) ↔ (𝐶 +o suc 𝐴) ⊆ suc (𝐶 +o 𝑦))) | 
| 28 | 24, 27 | imbitrrid 246 | . . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦))) | 
| 29 | 28 | ex 412 | . . . . . . . . . . . . . 14
⊢ (𝑦 ∈ On → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))) | 
| 30 | 29 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → (𝐶 ∈ On → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))) | 
| 31 | 30 | a2d 29 | . . . . . . . . . . . 12
⊢ (((𝑦 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑦) → ((𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦)) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o suc 𝑦)))) | 
| 32 |  | sucssel 6478 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ On → (suc 𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥)) | 
| 33 | 7, 32 | sylbir 235 | . . . . . . . . . . . . . . . . . . 19
⊢ (suc
𝐴 ∈ On → (suc
𝐴 ⊆ 𝑥 → 𝐴 ∈ 𝑥)) | 
| 34 |  | limsuc 7871 | . . . . . . . . . . . . . . . . . . . 20
⊢ (Lim
𝑥 → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥)) | 
| 35 | 34 | biimpd 229 | . . . . . . . . . . . . . . . . . . 19
⊢ (Lim
𝑥 → (𝐴 ∈ 𝑥 → suc 𝐴 ∈ 𝑥)) | 
| 36 | 33, 35 | sylan9r 508 | . . . . . . . . . . . . . . . . . 18
⊢ ((Lim
𝑥 ∧ suc 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝑥 → suc 𝐴 ∈ 𝑥)) | 
| 37 | 36 | imp 406 | . . . . . . . . . . . . . . . . 17
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → suc 𝐴 ∈ 𝑥) | 
| 38 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = suc 𝐴 → (𝐶 +o 𝑦) = (𝐶 +o suc 𝐴)) | 
| 39 | 38 | ssiun2s 5047 | . . . . . . . . . . . . . . . . 17
⊢ (suc
𝐴 ∈ 𝑥 → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 40 | 37, 39 | syl 17 | . . . . . . . . . . . . . . . 16
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 41 | 40 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 42 |  | vex 3483 | . . . . . . . . . . . . . . . . . . 19
⊢ 𝑥 ∈ V | 
| 43 |  | oalim 8571 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝐶 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 44 | 42, 43 | mpanr1 703 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐶 ∈ On ∧ Lim 𝑥) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 45 | 44 | ancoms 458 | . . . . . . . . . . . . . . . . 17
⊢ ((Lim
𝑥 ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 46 | 45 | adantlr 715 | . . . . . . . . . . . . . . . 16
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 47 | 46 | adantlr 715 | . . . . . . . . . . . . . . 15
⊢ ((((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐶 +o 𝑦)) | 
| 48 | 41, 47 | sseqtrrd 4020 | . . . . . . . . . . . . . 14
⊢ ((((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) ∧ 𝐶 ∈ On) → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)) | 
| 49 | 48 | ex 412 | . . . . . . . . . . . . 13
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥))) | 
| 50 | 49 | a1d 25 | . . . . . . . . . . . 12
⊢ (((Lim
𝑥 ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (suc 𝐴 ⊆ 𝑦 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑦))) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝑥)))) | 
| 51 | 10, 13, 16, 19, 21, 31, 50 | tfindsg 7883 | . . . . . . . . . . 11
⊢ (((𝐵 ∈ On ∧ suc 𝐴 ∈ On) ∧ suc 𝐴 ⊆ 𝐵) → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))) | 
| 52 | 51 | exp31 419 | . . . . . . . . . 10
⊢ (𝐵 ∈ On → (suc 𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))) | 
| 53 | 7, 52 | biimtrid 242 | . . . . . . . . 9
⊢ (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 ∈ On → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))) | 
| 54 | 53 | com4r 94 | . . . . . . . 8
⊢ (𝐶 ∈ On → (𝐵 ∈ On → (𝐴 ∈ On → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))))) | 
| 55 | 54 | imp31 417 | . . . . . . 7
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (suc 𝐴 ⊆ 𝐵 → (𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵))) | 
| 56 |  | oasuc 8563 | . . . . . . . . . 10
⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o suc 𝐴) = suc (𝐶 +o 𝐴)) | 
| 57 | 56 | sseq1d 4014 | . . . . . . . . 9
⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) ↔ suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) | 
| 58 |  | ovex 7465 | . . . . . . . . . 10
⊢ (𝐶 +o 𝐴) ∈ V | 
| 59 |  | sucssel 6478 | . . . . . . . . . 10
⊢ ((𝐶 +o 𝐴) ∈ V → (suc (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | 
| 60 | 58, 59 | ax-mp 5 | . . . . . . . . 9
⊢ (suc
(𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)) | 
| 61 | 57, 60 | biimtrdi 253 | . . . . . . . 8
⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | 
| 62 | 61 | adantlr 715 | . . . . . . 7
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → ((𝐶 +o suc 𝐴) ⊆ (𝐶 +o 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | 
| 63 | 6, 55, 62 | 3syld 60 | . . . . . 6
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | 
| 64 | 63 | imp 406 | . . . . 5
⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)) | 
| 65 | 64 | an32s 652 | . . . 4
⊢ ((((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)) | 
| 66 | 2, 65 | mpdan 687 | . . 3
⊢ (((𝐶 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)) | 
| 67 | 66 | ex 412 | . 2
⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | 
| 68 | 67 | ancoms 458 | 1
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) |