Proof of Theorem omltoe
Step | Hyp | Ref
| Expression |
1 | | simpr 484 |
. . . . . 6
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On) |
2 | 1 | adantr 480 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On) |
3 | | oe2 42759 |
. . . . 5
⊢ (𝐵 ∈ On → (𝐵 ·o 𝐵) = (𝐵 ↑o
2o)) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐵) = (𝐵 ↑o
2o)) |
5 | | 2on 8494 |
. . . . . . . . 9
⊢
2o ∈ On |
6 | 5 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o
∈ On) |
7 | | simpl 482 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) |
8 | 6, 7, 1 | 3jca 1126 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
(2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On)) |
9 | 8 | adantr 480 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧
𝐴 ∈ On ∧ 𝐵 ∈ On)) |
10 | | simpr 484 |
. . . . . . . . 9
⊢
((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
11 | 10 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ 𝐵) |
12 | 11 | ne0d 4331 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐵 ≠ ∅) |
13 | | on0eln0 6419 |
. . . . . . . 8
⊢ (𝐵 ∈ On → (∅
∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
14 | 2, 13 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
15 | 12, 14 | mpbird 257 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ∅ ∈ 𝐵) |
16 | 9, 15 | jca 511 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ((2o ∈ On ∧
𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈
𝐵)) |
17 | | df-2o 8481 |
. . . . . . 7
⊢
2o = suc 1o |
18 | 17 | a1i 11 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc
1o) |
19 | | simpl 482 |
. . . . . . . . 9
⊢
((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 1o ∈ 𝐴) |
20 | 19 | adantl 481 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴) |
21 | | eloni 6373 |
. . . . . . . . . 10
⊢ (𝐴 ∈ On → Ord 𝐴) |
22 | 21 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴) |
23 | 22 | adantr 480 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴) |
24 | 20, 23 | jca 511 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴)) |
25 | | ordelsuc 7817 |
. . . . . . . 8
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o
⊆ 𝐴)) |
26 | 25 | biimpd 228 |
. . . . . . 7
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o
⊆ 𝐴)) |
27 | 24, 20, 26 | sylc 65 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴) |
28 | 18, 27 | eqsstrd 4016 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴) |
29 | | oewordi 8605 |
. . . . 5
⊢
(((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (2o ⊆
𝐴 → (𝐵 ↑o 2o) ⊆
(𝐵 ↑o 𝐴))) |
30 | 16, 28, 29 | sylc 65 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ↑o 2o) ⊆
(𝐵 ↑o 𝐴)) |
31 | 4, 30 | eqsstrd 4016 |
. . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐵) ⊆ (𝐵 ↑o 𝐴)) |
32 | 2, 2, 15 | jca31 514 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵)) |
33 | | omordi 8580 |
. . . 4
⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈
𝐵) → (𝐴 ∈ 𝐵 → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵))) |
34 | 32, 11, 33 | sylc 65 |
. . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵)) |
35 | 31, 34 | sseldd 3979 |
. 2
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴)) |
36 | 35 | ex 412 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
((1o ∈ 𝐴
∧ 𝐴 ∈ 𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴))) |