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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 ∧ ¬ ∅ ∈ 𝐴 ) ↔ ( CoElEqvRel 𝐴 ∧ ( ∪ 𝐴 / ∼ 𝐴 ) = 𝐴 ) )
2		dfmembpart2	⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴 ) )
3		dfcomember3	⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( ∪ 𝐴 / ∼ 𝐴 ) = 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴 )