Metamath Proof Explorer


Theorem elcoeleqvrelsrel

Description: For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023)

Ref Expression
Assertion elcoeleqvrelsrel A V A CoElEqvRels CoElEqvRel A

Proof

Step Hyp Ref Expression
1 elcoeleqvrels A V A CoElEqvRels E -1 A EqvRels
2 1cosscnvepresex A V E -1 A V
3 eleqvrelsrel E -1 A V E -1 A EqvRels EqvRel E -1 A
4 2 3 syl A V E -1 A EqvRels EqvRel E -1 A
5 1 4 bitrd A V A CoElEqvRels EqvRel E -1 A
6 df-coeleqvrel CoElEqvRel A EqvRel E -1 A
7 5 6 bitr4di A V A CoElEqvRels CoElEqvRel A