Metamath Proof Explorer

Theorem cpet2

Description: The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet . Together with cpet , mpet mpet2 , this is what we used to think of as the partition equivalence theorem (but cf. pet2 with general R ). (Contributed by Peter Mazsa, 30-Dec-2024)

Ref Expression
Assertion cpet2 ⊢ ElDisj A ∧ ¬ ∅ ∈ A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 eldisjn0elb ⊢ ElDisj A ∧ ¬ ∅ ∈ A ↔ Disj E -1 ↾ A ∧ dom ⁡ E -1 ↾ A / E -1 ↾ A = A
2 eqvrelqseqdisj3 ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A → Disj E -1 ↾ A
3 2 petlem ⊢ Disj E -1 ↾ A ∧ dom ⁡ E -1 ↾ A / E -1 ↾ A = A ↔ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A
4 eqvreldmqs2 ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A
5 1 3 4 3bitri ⊢ ElDisj A ∧ ¬ ∅ ∈ A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A