Theorem infinf

Description: Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017)

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

Proof

Step	Hyp	Ref	Expression
1		isfinite	⊢ ( 𝐴 ∈ Fin ↔ 𝐴 ≺ ω )
2	1	notbii	⊢ ( ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω )
3		omex	⊢ ω ∈ V
4		domtri	⊢ ( ( ω ∈ V ∧ 𝐴 ∈ 𝐵 ) → ( ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω ) )
5	3 4	mpan	⊢ ( 𝐴 ∈ 𝐵 → ( ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω ) )
6	2 5	bitr4id	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴 ) )