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Metamath Proof Explorer

Theorem pet02

Description: Class A is a partition by the null class if and only if the cosets by the null class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021)

Ref Expression
Assertion pet02 ⊢ Disj ∅ ∧ dom ⁡ ∅ / ∅ = A ↔ EqvRel ≀ ∅ ∧ dom ⁡ ≀ ∅ / ≀ ∅ = A

Proof

Step Hyp Ref Expression
1 disjALTV0 ⊢ Disj ∅
2 1 petlemi ⊢ Disj ∅ ∧ dom ⁡ ∅ / ∅ = A ↔ EqvRel ≀ ∅ ∧ dom ⁡ ≀ ∅ / ≀ ∅ = A