Step |
Hyp |
Ref |
Expression |
1 |
|
oa0 |
⊢ ( 𝐴 ∈ On → ( 𝐴 +o ∅ ) = 𝐴 ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ∅ ) = 𝐴 ) |
3 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
4 |
|
0elon |
⊢ ∅ ∈ On |
5 |
|
oaword |
⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) ) |
6 |
5
|
3com13 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ ∅ ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) ) |
7 |
4 6
|
mp3an3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ⊆ 𝐵 ↔ ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) ) |
8 |
3 7
|
mpbii |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o ∅ ) ⊆ ( 𝐴 +o 𝐵 ) ) |
9 |
2 8
|
eqsstrrd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐴 +o 𝐵 ) ) |