Theorem sucdom

Description: Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013)

	Ref	Expression
Assertion	sucdom	$$⊢A\in \mathrm{\omega }\to \left(A\prec B\leftrightarrow \mathrm{suc}A\preccurlyeq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sucdom2	$$⊢A\prec B\to \mathrm{suc}A\preccurlyeq B$$
2		php4	$$⊢A\in \mathrm{\omega }\to A\prec \mathrm{suc}A$$
3		sdomdomtr	$$⊢\left(A\prec \mathrm{suc}A\wedge \mathrm{suc}A\preccurlyeq B\right)\to A\prec B$$
4	3	ex	$$⊢A\prec \mathrm{suc}A\to \left(\mathrm{suc}A\preccurlyeq B\to A\prec B\right)$$
5	2 4	syl	$$⊢A\in \mathrm{\omega }\to \left(\mathrm{suc}A\preccurlyeq B\to A\prec B\right)$$
6	1 5	impbid2	$$⊢A\in \mathrm{\omega }\to \left(A\prec B\leftrightarrow \mathrm{suc}A\preccurlyeq B\right)$$