Step |
Hyp |
Ref |
Expression |
1 |
|
alephord3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ↔ ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ) ) |
2 |
|
alephord3 |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐵 ⊆ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ⊆ ( ℵ ‘ 𝐴 ) ) ) |
3 |
2
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ⊆ 𝐴 ↔ ( ℵ ‘ 𝐵 ) ⊆ ( ℵ ‘ 𝐴 ) ) ) |
4 |
1 3
|
anbi12d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) ↔ ( ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ∧ ( ℵ ‘ 𝐵 ) ⊆ ( ℵ ‘ 𝐴 ) ) ) ) |
5 |
|
eqss |
⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) ) |
6 |
|
eqss |
⊢ ( ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) ↔ ( ( ℵ ‘ 𝐴 ) ⊆ ( ℵ ‘ 𝐵 ) ∧ ( ℵ ‘ 𝐵 ) ⊆ ( ℵ ‘ 𝐴 ) ) ) |
7 |
4 5 6
|
3bitr4g |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 = 𝐵 ↔ ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) ) ) |
8 |
7
|
bicomd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |