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Metamath Proof Explorer

Theorem mpet

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets . Member partition and comember equivalence relation are the same (or: each element of A have equivalent comembers if and only if A is a member partition). Together with mpet2 , mpet3 , and with the conventional cpet and cpet2 , this is what we used to think of as the partition equivalence theorem (but cf. pet2 with general R ). (Contributed by Peter Mazsa, 24-Sep-2021)

		Ref	Expression
	Assertion	mpet	\|- ( MembPart A <-> CoMembEr A )

Proof

Step	Hyp	Ref	Expression
1		mpet3	\|- ( ( ElDisj A /\ -. (/) e. A ) <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) )
2		dfmembpart2	\|- ( MembPart A <-> ( ElDisj A /\ -. (/) e. A ) )
3		dfcomember3	\|- ( CoMembEr A <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) )
4	1 2 3	3bitr4i	\|- ( MembPart A <-> CoMembEr A )