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	$$⊢A\in B\to \left(\neg A\in \mathrm{Fin}\leftrightarrow \mathrm{\omega }\preccurlyeq A\right)$$

Proof

Step	Hyp	Ref	Expression
1		omex	$$⊢\mathrm{\omega }\in \mathrm{V}$$
2		infinfg	$$⊢\left(\mathrm{\omega }\in \mathrm{V}\wedge A\in B\right)\to \left(\neg A\in \mathrm{Fin}\leftrightarrow \mathrm{\omega }\preccurlyeq A\right)$$
3	1 2	mpan	$$⊢A\in B\to \left(\neg A\in \mathrm{Fin}\leftrightarrow \mathrm{\omega }\preccurlyeq A\right)$$