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Metamath Proof Explorer

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 ) )