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Metamath Proof Explorer

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 ∧ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 dfmembpart2 ⊢ MembPart A ↔ ElDisj A ∧ ¬ ∅ ∈ A
2 cpet2 ⊢ ElDisj A ∧ ¬ ∅ ∈ A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A
3 1 2 bitri ⊢ MembPart A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A