Step |
Hyp |
Ref |
Expression |
1 |
|
idn1 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ) |
2 |
|
simpl |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → Ord 𝐴 ) |
3 |
1 2
|
e1a |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ Ord 𝐴 ) |
4 |
|
ordtr |
⊢ ( Ord 𝐴 → Tr 𝐴 ) |
5 |
3 4
|
e1a |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ Tr 𝐴 ) |
6 |
|
dford2 |
⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) ) |
7 |
6
|
simprbi |
⊢ ( Ord 𝐴 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) |
8 |
3 7
|
e1a |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) |
9 |
|
3orcomb |
⊢ ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
10 |
9
|
ax-gen |
⊢ ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
11 |
|
alral |
⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) → ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
12 |
10 11
|
e0a |
⊢ ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
13 |
|
ralbi |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) → ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
14 |
12 13
|
e0a |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
15 |
14
|
ax-gen |
⊢ ∀ 𝑥 ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
16 |
|
alral |
⊢ ( ∀ 𝑥 ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) → ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
17 |
15 16
|
e0a |
⊢ ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
18 |
|
ralbi |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) ) |
19 |
17 18
|
e0a |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
20 |
8 19
|
e1bi |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ) |
21 |
|
simpr |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) |
22 |
1 21
|
e1a |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ 𝐵 ∈ 𝐴 ) |
23 |
|
tratrb |
⊢ ( ( Tr 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) ∧ 𝐵 ∈ 𝐴 ) → Tr 𝐵 ) |
24 |
23
|
3exp |
⊢ ( Tr 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦 ) → ( 𝐵 ∈ 𝐴 → Tr 𝐵 ) ) ) |
25 |
5 20 22 24
|
e111 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ Tr 𝐵 ) |
26 |
|
trss |
⊢ ( Tr 𝐴 → ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴 ) ) |
27 |
5 22 26
|
e11 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ 𝐵 ⊆ 𝐴 ) |
28 |
|
ssralv2 |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) ) |
29 |
28
|
ex |
⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐵 ⊆ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) ) ) |
30 |
27 27 8 29
|
e111 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) |
31 |
|
dford2 |
⊢ ( Ord 𝐵 ↔ ( Tr 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) ) ) |
32 |
31
|
simplbi2 |
⊢ ( Tr 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 ) → Ord 𝐵 ) ) |
33 |
25 30 32
|
e11 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) ▶ Ord 𝐵 ) |
34 |
33
|
in1 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → Ord 𝐵 ) |