Theorem xp0

Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of Monk1 p. 37. (Contributed by NM, 12-Apr-2004)

	Ref	Expression
Assertion	xp0	⊢ ( 𝐴 × ∅ ) = ∅

Proof

Step	Hyp	Ref	Expression
1		0xp	⊢ ( ∅ × 𝐴 ) = ∅
2	1	cnveqi	⊢ ◡ ( ∅ × 𝐴 ) = ◡ ∅
3		cnvxp	⊢ ◡ ( ∅ × 𝐴 ) = ( 𝐴 × ∅ )
4		cnv0	⊢ ◡ ∅ = ∅
5	2 3 4	3eqtr3i	⊢ ( 𝐴 × ∅ ) = ∅