Theorem aleph11

Description: The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004)

	Ref	Expression
Assertion	aleph11	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( ℵ ‘ 𝐴 ) = ( ℵ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

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 ) )