Theorem dfcoeleqvrels

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 , eqvrelcoss3 and eqvrelcoss4 when needed. (Contributed by Peter Mazsa, 28-Nov-2022)

	Ref	Expression
Assertion	dfcoeleqvrels	|- CoElEqvRels = { a | ~ a e. EqvRels }

Proof

Step	Hyp	Ref	Expression
1		df-coeleqvrels	|- CoElEqvRels = { a | ,~ ( `' _E |` a ) e. EqvRels }
2		df-coels	|- ~ a = ,~ ( `' _E |` a )
3	2	eleq1i	|- ( ~ a e. EqvRels <-> ,~ ( `' _E |` a ) e. EqvRels )
4	3	abbii	|- { a | ~ a e. EqvRels } = { a | ,~ ( `' _E |` a ) e. EqvRels }
5	1 4	eqtr4i	|- CoElEqvRels = { a | ~ a e. EqvRels }