Metamath Proof Explorer


Theorem unnum

Description: The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015)

Ref Expression
Assertion unnum ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴𝐵 ) ∈ dom card )

Proof

Step Hyp Ref Expression
1 djunum ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴𝐵 ) ∈ dom card )
2 undjudom ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴𝐵 ) ≼ ( 𝐴𝐵 ) )
3 numdom ( ( ( 𝐴𝐵 ) ∈ dom card ∧ ( 𝐴𝐵 ) ≼ ( 𝐴𝐵 ) ) → ( 𝐴𝐵 ) ∈ dom card )
4 1 2 3 syl2anc ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴𝐵 ) ∈ dom card )