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 )