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

Proof

Step	Hyp	Ref	Expression
1		disjsn	⊢ ( ( 𝐴 ∩ { 𝑋 } ) = ∅ ↔ ¬ 𝑋 ∈ 𝐴 )
2		xpima1	⊢ ( ( 𝐴 ∩ { 𝑋 } ) = ∅ → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = ∅ )
3	1 2	sylbir	⊢ ( ¬ 𝑋 ∈ 𝐴 → ( ( 𝐴 × 𝐵 ) “ { 𝑋 } ) = ∅ )