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	$$⊢MembPartA\leftrightarrow CoMembErA$$

Proof

Step	Hyp	Ref	Expression
1		mpet3	$$⊢\left(ElDisjA\wedge \neg \varnothing \in A\right)\leftrightarrow \left(CoElEqvRelA\wedge \bigcup A/\sim A=A\right)$$
2		dfmembpart2	$$⊢MembPartA\leftrightarrow \left(ElDisjA\wedge \neg \varnothing \in A\right)$$
3		dfcomember3	$$⊢CoMembErA\leftrightarrow \left(CoElEqvRelA\wedge \bigcup A/\sim A=A\right)$$
4	1 2 3	3bitr4i	$$⊢MembPartA\leftrightarrow CoMembErA$$