Theorem infinf

Description: Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017) (Proof shortened by Scott Fenton, 20-Feb-2026)

	Ref	Expression
Assertion	infinf	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		omex	⊢ ω ∈ V
2		infinfg	⊢ ( ( ω ∈ V ∧ 𝐴 ∈ 𝐵 ) → ( ¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴 ) )