Theorem scottn0f

Description: A version of scott0f with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018)

	Ref	Expression
Hypotheses	scottn0f.1	|- F/_ y A
	scottn0f.2	|- F/_ x A
Assertion	scottn0f	|- ( A =/= (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } =/= (/) )

Proof

Step	Hyp	Ref	Expression
1		scottn0f.1	|- F/_ y A
2		scottn0f.2	|- F/_ x A
3	1 2	scott0f	|- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )
4	3	necon3bii	|- ( A =/= (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } =/= (/) )