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	⊢ ( ¬ 𝑋 ∈ 𝐴 → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		bj-xpimasn	⊢ ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = if ( 𝑋 ∈ 𝐴 , 𝐵 , ∅ )
2		iffalse	⊢ ( ¬ 𝑋 ∈ 𝐴 → if ( 𝑋 ∈ 𝐴 , 𝐵 , ∅ ) = ∅ )
3	1 2	syl5eq	⊢ ( ¬ 𝑋 ∈ 𝐴 → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = ∅ )