Theorem enfi

Description: Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008)

	Ref	Expression
Assertion	enfi	⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ Fin ↔ 𝐵 ∈ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		enen1	⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥 ) )
2	1	rexbidv	⊢ ( 𝐴 ≈ 𝐵 → ( ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃ 𝑥 ∈ ω 𝐵 ≈ 𝑥 ) )
3		isfi	⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑥 ∈ ω 𝐴 ≈ 𝑥 )
4		isfi	⊢ ( 𝐵 ∈ Fin ↔ ∃ 𝑥 ∈ ω 𝐵 ≈ 𝑥 )
5	2 3 4	3bitr4g	⊢ ( 𝐴 ≈ 𝐵 → ( 𝐴 ∈ Fin ↔ 𝐵 ∈ Fin ) )