Metamath Proof Explorer


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 ) )