Step |
Hyp |
Ref |
Expression |
1 |
|
0elon |
⊢ ∅ ∈ On |
2 |
|
oaord |
⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) ) |
3 |
1 2
|
mp3an1 |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) ) |
4 |
|
oa0 |
⊢ ( 𝐴 ∈ On → ( 𝐴 +o ∅ ) = 𝐴 ) |
5 |
4
|
adantl |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐴 +o ∅ ) = 𝐴 ) |
6 |
5
|
eleq1d |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) ) |
7 |
3 6
|
bitrd |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) ) |
8 |
7
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐴 ∈ ( 𝐴 +o 𝐵 ) ) ) |