Theorem peano2b

Description: A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995)

	Ref	Expression
Assertion	peano2b	$$⊢A\in \mathrm{\omega }\leftrightarrow \mathrm{suc}A\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		limom	$$⊢\mathrm{Lim}\mathrm{\omega }$$
2		limsuc	$$⊢\mathrm{Lim}\mathrm{\omega }\to \left(A\in \mathrm{\omega }\leftrightarrow \mathrm{suc}A\in \mathrm{\omega }\right)$$
3	1 2	ax-mp	$$⊢A\in \mathrm{\omega }\leftrightarrow \mathrm{suc}A\in \mathrm{\omega }$$