Theorem pssnel

Description: A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996)

	Ref	Expression
Assertion	pssnel	$$⊢A\subset B\to \exists x\left(x\in B\wedge \neg x\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		pssdif	$$⊢A\subset B\to B\setminus A\ne \varnothing$$
2		n0	$$⊢B\setminus A\ne \varnothing \leftrightarrow \exists xx\in \left(B\setminus A\right)$$
3	1 2	sylib	$$⊢A\subset B\to \exists xx\in \left(B\setminus A\right)$$
4		eldif	$$⊢x\in \left(B\setminus A\right)\leftrightarrow \left(x\in B\wedge \neg x\in A\right)$$
5	4	exbii	$$⊢\exists xx\in \left(B\setminus A\right)\leftrightarrow \exists x\left(x\in B\wedge \neg x\in A\right)$$
6	3 5	sylib	$$⊢A\subset B\to \exists x\left(x\in B\wedge \neg x\in A\right)$$