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Metamath Proof Explorer

Theorem unfid

Description: The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses unfid.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
unfid.2 ⊢ ( 𝜑 → 𝐵 ∈ Fin )
Assertion unfid ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ Fin )

Proof

Step Hyp Ref Expression
1 unfid.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
2 unfid.2 ⊢ ( 𝜑 → 𝐵 ∈ Fin )
3 unfi ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( 𝐴 ∪ 𝐵 ) ∈ Fin )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ Fin )