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Metamath Proof Explorer

Theorem pet0

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 pet0 ⊢ ∅ Part A ↔ ≀ ∅ ErALTV A

Proof

Step Hyp Ref Expression
1 pet02 ⊢ Disj ∅ ∧ dom ⁡ ∅ / ∅ = A ↔ EqvRel ≀ ∅ ∧ dom ⁡ ≀ ∅ / ≀ ∅ = A
2 dfpart2 ⊢ ∅ Part A ↔ Disj ∅ ∧ dom ⁡ ∅ / ∅ = A
3 dferALTV2 ⊢ ≀ ∅ ErALTV A ↔ EqvRel ≀ ∅ ∧ dom ⁡ ≀ ∅ / ≀ ∅ = A
4 1 2 3 3bitr4i ⊢ ∅ Part A ↔ ≀ ∅ ErALTV A