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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 ) )