Theorem hashelne0d

Description: A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	hashelne0d.1	|- ( ph -> B e. A )
	hashelne0d.2	|- ( ph -> A e. V )
Assertion	hashelne0d	|- ( ph -> -. ( # ` A ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		hashelne0d.1	|- ( ph -> B e. A )
2		hashelne0d.2	|- ( ph -> A e. V )
3	1	ne0d	|- ( ph -> A =/= (/) )
4	3	neneqd	|- ( ph -> -. A = (/) )
5		hasheq0	|- ( A e. V -> ( ( # ` A ) = 0 <-> A = (/) ) )
6	2 5	syl	|- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
7	4 6	mtbird	|- ( ph -> -. ( # ` A ) = 0 )