Theorem bj-xpima1snALT

Description: Alternate proof of bj-xpima1sn . (Contributed by BJ, 6-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-xpima1snALT	$$⊢\neg X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		disjsn	$$⊢A\cap \left\{X\right\}=\varnothing \leftrightarrow \neg X\in A$$
2		xpima1	$$⊢A\cap \left\{X\right\}=\varnothing \to \left(A\times B\right)\left[\left\{X\right\}\right]=\varnothing$$
3	1 2	sylbir	$$⊢\neg X\in A\to \left(A\times B\right)\left[\left\{X\right\}\right]=\varnothing$$