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Metamath Proof Explorer

Theorem mpets2

Description: Member Partition-Equivalence Theorem with binary relations, cf. mpet2 . (Contributed by Peter Mazsa, 24-Sep-2021)

Ref Expression
Assertion mpets2 
|- ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) )

Proof

Step Hyp Ref Expression
1 mpet2 
 |-  ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A )
2 cnvepresex 
 |-  ( A e. V -> ( `' _E |` A ) e. _V )
3 brpartspart 
 |-  ( ( A e. V /\ ( `' _E |` A ) e. _V ) -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) )
4 2 3 mpdan 
 |-  ( A e. V -> ( ( `' _E |` A ) Parts A <-> ( `' _E |` A ) Part A ) )
5 1cosscnvepresex 
 |-  ( A e. V -> ,~ ( `' _E |` A ) e. _V )
6 brerser 
 |-  ( ( A e. V /\ ,~ ( `' _E |` A ) e. _V ) -> ( ,~ ( `' _E |` A ) Ers A <-> ,~ ( `' _E |` A ) ErALTV A ) )
7 5 6 mpdan 
 |-  ( A e. V -> ( ,~ ( `' _E |` A ) Ers A <-> ,~ ( `' _E |` A ) ErALTV A ) )
8 4 7 bibi12d 
 |-  ( A e. V -> ( ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) <-> ( ( `' _E |` A ) Part A <-> ,~ ( `' _E |` A ) ErALTV A ) ) )
9 1 8 mpbiri 
 |-  ( A e. V -> ( ( `' _E |` A ) Parts A <-> ,~ ( `' _E |` A ) Ers A ) )