Theorem eleldisjs

Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023)

	Ref	Expression
Assertion	eleldisjs	|- ( A e. V -> ( A e. ElDisjs <-> ( `' _E |` A ) e. Disjs ) )

Proof

Step	Hyp	Ref	Expression
1		reseq2	|- ( a = A -> ( `' _E |` a ) = ( `' _E |` A ) )
2	1	eleq1d	|- ( a = A -> ( ( `' _E |` a ) e. Disjs <-> ( `' _E |` A ) e. Disjs ) )
3		df-eldisjs	|- ElDisjs = { a | ( `' _E |` a ) e. Disjs }
4	2 3	elab2g	|- ( A e. V -> ( A e. ElDisjs <-> ( `' _E |` A ) e. Disjs ) )