Step |
Hyp |
Ref |
Expression |
1 |
|
alephord |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) ) ) |
2 |
|
alephon |
⊢ ( ℵ ‘ 𝐴 ) ∈ On |
3 |
|
alephon |
⊢ ( ℵ ‘ 𝐵 ) ∈ On |
4 |
|
onenon |
⊢ ( ( ℵ ‘ 𝐵 ) ∈ On → ( ℵ ‘ 𝐵 ) ∈ dom card ) |
5 |
3 4
|
ax-mp |
⊢ ( ℵ ‘ 𝐵 ) ∈ dom card |
6 |
|
cardsdomel |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ On ∧ ( ℵ ‘ 𝐵 ) ∈ dom card ) → ( ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) ↔ ( ℵ ‘ 𝐴 ) ∈ ( card ‘ ( ℵ ‘ 𝐵 ) ) ) ) |
7 |
2 5 6
|
mp2an |
⊢ ( ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) ↔ ( ℵ ‘ 𝐴 ) ∈ ( card ‘ ( ℵ ‘ 𝐵 ) ) ) |
8 |
|
alephcard |
⊢ ( card ‘ ( ℵ ‘ 𝐵 ) ) = ( ℵ ‘ 𝐵 ) |
9 |
8
|
eleq2i |
⊢ ( ( ℵ ‘ 𝐴 ) ∈ ( card ‘ ( ℵ ‘ 𝐵 ) ) ↔ ( ℵ ‘ 𝐴 ) ∈ ( ℵ ‘ 𝐵 ) ) |
10 |
7 9
|
bitri |
⊢ ( ( ℵ ‘ 𝐴 ) ≺ ( ℵ ‘ 𝐵 ) ↔ ( ℵ ‘ 𝐴 ) ∈ ( ℵ ‘ 𝐵 ) ) |
11 |
1 10
|
bitrdi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∈ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ∈ ( ℵ ‘ 𝐵 ) ) ) |