Theorem n0el3

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021)

	Ref	Expression
Assertion	n0el3	|- ( -. (/) e. A <-> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		n0elim	|- ( -. (/) e. A -> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A )
2		n0eldmqseq	|- ( ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A -> -. (/) e. A )
3	1 2	impbii	|- ( -. (/) e. A <-> ( dom ( `' _E |` A ) /. ( `' _E |` A ) ) = A )