Step |
Hyp |
Ref |
Expression |
1 |
|
om1r |
⊢ ( 𝐴 ∈ On → ( 1o ·o 𝐴 ) = 𝐴 ) |
2 |
1
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐵 ) → ( 1o ·o 𝐴 ) = 𝐴 ) |
3 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
4 |
|
ordgt0ge1 |
⊢ ( Ord 𝐵 → ( ∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵 ) ) |
5 |
4
|
biimpa |
⊢ ( ( Ord 𝐵 ∧ ∅ ∈ 𝐵 ) → 1o ⊆ 𝐵 ) |
6 |
3 5
|
sylan |
⊢ ( ( 𝐵 ∈ On ∧ ∅ ∈ 𝐵 ) → 1o ⊆ 𝐵 ) |
7 |
6
|
adantll |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐵 ) → 1o ⊆ 𝐵 ) |
8 |
|
1on |
⊢ 1o ∈ On |
9 |
|
omwordri |
⊢ ( ( 1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 1o ⊆ 𝐵 → ( 1o ·o 𝐴 ) ⊆ ( 𝐵 ·o 𝐴 ) ) ) |
10 |
8 9
|
mp3an1 |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 1o ⊆ 𝐵 → ( 1o ·o 𝐴 ) ⊆ ( 𝐵 ·o 𝐴 ) ) ) |
11 |
10
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 1o ⊆ 𝐵 → ( 1o ·o 𝐴 ) ⊆ ( 𝐵 ·o 𝐴 ) ) ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐵 ) → ( 1o ⊆ 𝐵 → ( 1o ·o 𝐴 ) ⊆ ( 𝐵 ·o 𝐴 ) ) ) |
13 |
7 12
|
mpd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐵 ) → ( 1o ·o 𝐴 ) ⊆ ( 𝐵 ·o 𝐴 ) ) |
14 |
2 13
|
eqsstrrd |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐵 ) → 𝐴 ⊆ ( 𝐵 ·o 𝐴 ) ) |