Theorem bj-xpima1sn

Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-xpima1sn	$$⊢\neg X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		bj-xpimasn	$$⊢\left(A\times B\right)\left[\left\{X\right\}\right]=if\left(X\in A,B,\varnothing \right)$$
2		iffalse	$$⊢\neg X\in A\to if\left(X\in A,B,\varnothing \right)=\varnothing$$
3	1 2	syl5eq	$$⊢\neg X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=\varnothing$$