Theorem cpet

Description: The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 . Cf. mpet , mpet2 and mpet3 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet and pet2 for Partition-Equivalence Theorem with general R . (Contributed by Peter Mazsa, 31-Dec-2024)

	Ref	Expression
Assertion	cpet	|- ( MembPart A <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		dfmembpart2	|- ( MembPart A <-> ( ElDisj A /\ -. (/) e. A ) )
2		cpet2	|- ( ( ElDisj A /\ -. (/) e. A ) <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) )
3	1 2	bitri	|- ( MembPart A <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) )