Theorem in0

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of Suppes p. 26. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	in0	⊢ ( 𝐴 ∩ ∅ ) = ∅

Proof

Step	Hyp	Ref	Expression
1		noel	⊢ ¬ 𝑥 ∈ ∅
2	1	bianfi	⊢ ( 𝑥 ∈ ∅ ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅ ) )
3	2	bicomi	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅ ) ↔ 𝑥 ∈ ∅ )
4	3	ineqri	⊢ ( 𝐴 ∩ ∅ ) = ∅