Step |
Hyp |
Ref |
Expression |
1 |
|
alephord2 |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐵 ∈ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) ) |
2 |
1
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) ) |
3 |
2
|
notbid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝐵 ∈ 𝐴 ↔ ¬ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) ) |
4 |
|
ontri1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴 ) ) |
5 |
|
alephon |
⊢ ( ℵ ‘ 𝐴 ) ∈ On |
6 |
|
alephon |
⊢ ( ℵ ‘ 𝐵 ) ∈ On |
7 |
|
ontri1 |
⊢ ( ( ( ℵ ‘ 𝐴 ) ∈ On ∧ ( ℵ ‘ 𝐵 ) ∈ On ) → ( ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ↔ ¬ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) ) |
8 |
5 6 7
|
mp2an |
⊢ ( ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ↔ ¬ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) |
9 |
8
|
a1i |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ↔ ¬ ( ℵ ‘ 𝐵 ) ∈ ( ℵ ‘ 𝐴 ) ) ) |
10 |
3 4 9
|
3bitr4d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ) ) |