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 ) → ( 𝐴 ∪ 𝐵 ) ∈ dom card ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djunum | ⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴 ⊔ 𝐵 ) ∈ dom card ) | |
2 | undjudom | ⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴 ∪ 𝐵 ) ≼ ( 𝐴 ⊔ 𝐵 ) ) | |
3 | numdom | ⊢ ( ( ( 𝐴 ⊔ 𝐵 ) ∈ dom card ∧ ( 𝐴 ∪ 𝐵 ) ≼ ( 𝐴 ⊔ 𝐵 ) ) → ( 𝐴 ∪ 𝐵 ) ∈ dom card ) | |
4 | 1 2 3 | syl2anc | ⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴 ∪ 𝐵 ) ∈ dom card ) |