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	$$⊢A\cap \varnothing =\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		noel	$$⊢\neg x\in \varnothing$$
2	1	bianfi	$$⊢x\in \varnothing \leftrightarrow \left(x\in A\wedge x\in \varnothing \right)$$
3	2	bicomi	$$⊢\left(x\in A\wedge x\in \varnothing \right)\leftrightarrow x\in \varnothing$$
4	3	ineqri	$$⊢A\cap \varnothing =\varnothing$$