Proof of Theorem onov0suclim
| Step | Hyp | Ref
| Expression |
| 1 | | eloni 6394 |
. . . 4
⊢ (𝐵 ∈ On → Ord 𝐵) |
| 2 | | orduniorsuc 7850 |
. . . . 5
⊢ (Ord
𝐵 → (𝐵 = ∪ 𝐵 ∨ 𝐵 = suc ∪ 𝐵)) |
| 3 | | unizlim 6507 |
. . . . . . 7
⊢ (Ord
𝐵 → (𝐵 = ∪ 𝐵 ↔ (𝐵 = ∅ ∨ Lim 𝐵))) |
| 4 | 3 | biimpd 229 |
. . . . . 6
⊢ (Ord
𝐵 → (𝐵 = ∪ 𝐵 → (𝐵 = ∅ ∨ Lim 𝐵))) |
| 5 | 4 | orim1d 968 |
. . . . 5
⊢ (Ord
𝐵 → ((𝐵 = ∪
𝐵 ∨ 𝐵 = suc ∪ 𝐵) → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc ∪ 𝐵))) |
| 6 | 2, 5 | mpd 15 |
. . . 4
⊢ (Ord
𝐵 → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc ∪ 𝐵)) |
| 7 | 1, 6 | syl 17 |
. . 3
⊢ (𝐵 ∈ On → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc ∪ 𝐵)) |
| 8 | 7 | adantl 481 |
. 2
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc ∪ 𝐵)) |
| 9 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = (𝐴 ⊗
∅)) |
| 10 | | onov0suclim.0 |
. . . . . . . . 9
⊢ (𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷) |
| 11 | 9, 10 | sylan9eqr 2799 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ⊗ 𝐵) = 𝐷) |
| 12 | 11 | ex 412 |
. . . . . . 7
⊢ (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷)) |
| 13 | 12 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷)) |
| 14 | | eloni 6394 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ On → Ord 𝐶) |
| 15 | | 0elsuc 7855 |
. . . . . . . . . . . . 13
⊢ (Ord
𝐶 → ∅ ∈ suc
𝐶) |
| 16 | 14, 15 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ On → ∅ ∈
suc 𝐶) |
| 17 | 16 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → ∅ ∈ suc 𝐶) |
| 18 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → 𝐵 = suc 𝐶) |
| 19 | 17, 18 | eleqtrrd 2844 |
. . . . . . . . . 10
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → ∅ ∈ 𝐵) |
| 20 | | n0i 4340 |
. . . . . . . . . 10
⊢ (∅
∈ 𝐵 → ¬ 𝐵 = ∅) |
| 21 | 19, 20 | syl 17 |
. . . . . . . . 9
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → ¬ 𝐵 = ∅) |
| 22 | 21 | pm2.21d 121 |
. . . . . . . 8
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 23 | 22 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 24 | 23 | impancom 451 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 25 | | nlim0 6443 |
. . . . . . . . 9
⊢ ¬
Lim ∅ |
| 26 | | limeq 6396 |
. . . . . . . . 9
⊢ (𝐵 = ∅ → (Lim 𝐵 ↔ Lim
∅)) |
| 27 | 25, 26 | mtbiri 327 |
. . . . . . . 8
⊢ (𝐵 = ∅ → ¬ Lim
𝐵) |
| 28 | 27 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ¬ Lim
𝐵) |
| 29 | 28 | pm2.21d 121 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)) |
| 30 | 13, 24, 29 | 3jca 1129 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = ∅) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹))) |
| 31 | 30 | ex 412 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))) |
| 32 | 27 | con2i 139 |
. . . . . . . 8
⊢ (Lim
𝐵 → ¬ 𝐵 = ∅) |
| 33 | 32 | adantl 481 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ¬ 𝐵 = ∅) |
| 34 | 33 | pm2.21d 121 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷)) |
| 35 | | limeq 6396 |
. . . . . . . . . . . 12
⊢ (𝐵 = suc 𝐶 → (Lim 𝐵 ↔ Lim suc 𝐶)) |
| 36 | 35 | notbid 318 |
. . . . . . . . . . 11
⊢ (𝐵 = suc 𝐶 → (¬ Lim 𝐵 ↔ ¬ Lim suc 𝐶)) |
| 37 | 36 | biimprd 248 |
. . . . . . . . . 10
⊢ (𝐵 = suc 𝐶 → (¬ Lim suc 𝐶 → ¬ Lim 𝐵)) |
| 38 | | nlimsucg 7863 |
. . . . . . . . . 10
⊢ (𝐶 ∈ On → ¬ Lim suc
𝐶) |
| 39 | 37, 38 | impel 505 |
. . . . . . . . 9
⊢ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → ¬ Lim 𝐵) |
| 40 | 39 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → ¬ Lim 𝐵) |
| 41 | 40 | pm2.21d 121 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 42 | 41 | impancom 451 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 43 | | onov0suclim.lim |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹) |
| 44 | 43 | a1d 25 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)) |
| 45 | 34, 42, 44 | 3jca 1129 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹))) |
| 46 | 45 | ex 412 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Lim 𝐵 → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))) |
| 47 | 31, 46 | jaod 860 |
. . 3
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ ∨ Lim 𝐵) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))) |
| 48 | | 1n0 8526 |
. . . . . . . . 9
⊢
1o ≠ ∅ |
| 49 | | necom 2994 |
. . . . . . . . . . 11
⊢
(1o ≠ ∅ ↔ ∅ ≠
1o) |
| 50 | | df-1o 8506 |
. . . . . . . . . . . . 13
⊢
1o = suc ∅ |
| 51 | | uni0 4935 |
. . . . . . . . . . . . . 14
⊢ ∪ ∅ = ∅ |
| 52 | | suceq 6450 |
. . . . . . . . . . . . . 14
⊢ (∪ ∅ = ∅ → suc ∪ ∅ = suc ∅) |
| 53 | 51, 52 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ suc ∪ ∅ = suc ∅ |
| 54 | 50, 53 | eqtr4i 2768 |
. . . . . . . . . . . 12
⊢
1o = suc ∪
∅ |
| 55 | 54 | neeq2i 3006 |
. . . . . . . . . . 11
⊢ (∅
≠ 1o ↔ ∅ ≠ suc ∪
∅) |
| 56 | | df-ne 2941 |
. . . . . . . . . . 11
⊢ (∅
≠ suc ∪ ∅ ↔ ¬ ∅ = suc ∪ ∅) |
| 57 | 49, 55, 56 | 3bitri 297 |
. . . . . . . . . 10
⊢
(1o ≠ ∅ ↔ ¬ ∅ = suc ∪ ∅) |
| 58 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝐵 = ∅ → 𝐵 = ∅) |
| 59 | | unieq 4918 |
. . . . . . . . . . . . 13
⊢ (𝐵 = ∅ → ∪ 𝐵 =
∪ ∅) |
| 60 | | suceq 6450 |
. . . . . . . . . . . . 13
⊢ (∪ 𝐵 =
∪ ∅ → suc ∪
𝐵 = suc ∪ ∅) |
| 61 | 59, 60 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐵 = ∅ → suc ∪ 𝐵 =
suc ∪ ∅) |
| 62 | 58, 61 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝐵 = ∅ → (𝐵 = suc ∪ 𝐵
↔ ∅ = suc ∪ ∅)) |
| 63 | 62 | notbid 318 |
. . . . . . . . . 10
⊢ (𝐵 = ∅ → (¬ 𝐵 = suc ∪ 𝐵
↔ ¬ ∅ = suc ∪
∅)) |
| 64 | 57, 63 | bitr4id 290 |
. . . . . . . . 9
⊢ (𝐵 = ∅ → (1o
≠ ∅ ↔ ¬ 𝐵
= suc ∪ 𝐵)) |
| 65 | 48, 64 | mpbii 233 |
. . . . . . . 8
⊢ (𝐵 = ∅ → ¬ 𝐵 = suc ∪ 𝐵) |
| 66 | 65 | con2i 139 |
. . . . . . 7
⊢ (𝐵 = suc ∪ 𝐵
→ ¬ 𝐵 =
∅) |
| 67 | 66 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ ¬ 𝐵 =
∅) |
| 68 | 67 | pm2.21d 121 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ (𝐵 = ∅ →
(𝐴 ⊗ 𝐵) = 𝐷)) |
| 69 | | simprl 771 |
. . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → 𝐵 = suc 𝐶) |
| 70 | 69 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → (𝐴 ⊗ 𝐵) = (𝐴 ⊗ suc 𝐶)) |
| 71 | | onov0suclim.suc |
. . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸) |
| 72 | 71 | adantrl 716 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → (𝐴 ⊗ suc 𝐶) = 𝐸) |
| 73 | 70, 72 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ (𝐵 = suc 𝐶 ∧ 𝐶 ∈ On)) → (𝐴 ⊗ 𝐵) = 𝐸) |
| 74 | 73 | ex 412 |
. . . . . 6
⊢ (𝐴 ∈ On → ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 75 | 74 | ad2antrr 726 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸)) |
| 76 | | onuni 7808 |
. . . . . . . . 9
⊢ (𝐵 ∈ On → ∪ 𝐵
∈ On) |
| 77 | | nlimsucg 7863 |
. . . . . . . . 9
⊢ (∪ 𝐵
∈ On → ¬ Lim suc ∪ 𝐵) |
| 78 | 76, 77 | syl 17 |
. . . . . . . 8
⊢ (𝐵 ∈ On → ¬ Lim suc
∪ 𝐵) |
| 79 | | limeq 6396 |
. . . . . . . . . 10
⊢ (𝐵 = suc ∪ 𝐵
→ (Lim 𝐵 ↔ Lim
suc ∪ 𝐵)) |
| 80 | 79 | notbid 318 |
. . . . . . . . 9
⊢ (𝐵 = suc ∪ 𝐵
→ (¬ Lim 𝐵 ↔
¬ Lim suc ∪ 𝐵)) |
| 81 | 80 | biimprd 248 |
. . . . . . . 8
⊢ (𝐵 = suc ∪ 𝐵
→ (¬ Lim suc ∪ 𝐵 → ¬ Lim 𝐵)) |
| 82 | 78, 81 | mpan9 506 |
. . . . . . 7
⊢ ((𝐵 ∈ On ∧ 𝐵 = suc ∪ 𝐵)
→ ¬ Lim 𝐵) |
| 83 | 82 | adantll 714 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ ¬ Lim 𝐵) |
| 84 | 83 | pm2.21d 121 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)) |
| 85 | 68, 75, 84 | 3jca 1129 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐵 = suc ∪ 𝐵)
→ ((𝐵 = ∅ →
(𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹))) |
| 86 | 85 | ex 412 |
. . 3
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = suc ∪ 𝐵
→ ((𝐵 = ∅ →
(𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))) |
| 87 | 47, 86 | jaod 860 |
. 2
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝐵 = ∅ ∨ Lim 𝐵) ∨ 𝐵 = suc ∪ 𝐵) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))) |
| 88 | 8, 87 | mpd 15 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹))) |