Theorem opnz

Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opnz	|- ( <. A , B >. =/= (/) <-> ( A e. _V /\ B e. _V ) )

Proof

Step	Hyp	Ref	Expression
1		opprc	|- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
2	1	necon1ai	|- ( <. A , B >. =/= (/) -> ( A e. _V /\ B e. _V ) )
3		dfopg	|- ( ( A e. _V /\ B e. _V ) -> <. A , B >. = { { A } , { A , B } } )
4		snex	|- { A } e. _V
5	4	prnz	|- { { A } , { A , B } } =/= (/)
6	5	a1i	|- ( ( A e. _V /\ B e. _V ) -> { { A } , { A , B } } =/= (/) )
7	3 6	eqnetrd	|- ( ( A e. _V /\ B e. _V ) -> <. A , B >. =/= (/) )
8	2 7	impbii	|- ( <. A , B >. =/= (/) <-> ( A e. _V /\ B e. _V ) )