Theorem difxp2

Description: Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013) (Revised by Mario Carneiro, 26-Jun-2014)

	Ref	Expression
Assertion	difxp2	⊢ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 × 𝐵 ) ∖ ( 𝐴 × 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		difxp	⊢ ( ( 𝐴 × 𝐵 ) ∖ ( 𝐴 × 𝐶 ) ) = ( ( ( 𝐴 ∖ 𝐴 ) × 𝐵 ) ∪ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) )
2		difid	⊢ ( 𝐴 ∖ 𝐴 ) = ∅
3	2	xpeq1i	⊢ ( ( 𝐴 ∖ 𝐴 ) × 𝐵 ) = ( ∅ × 𝐵 )
4		0xp	⊢ ( ∅ × 𝐵 ) = ∅
5	3 4	eqtri	⊢ ( ( 𝐴 ∖ 𝐴 ) × 𝐵 ) = ∅
6	5	uneq1i	⊢ ( ( ( 𝐴 ∖ 𝐴 ) × 𝐵 ) ∪ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ) = ( ∅ ∪ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) )
7		uncom	⊢ ( ∅ ∪ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ) = ( ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ∪ ∅ )
8		un0	⊢ ( ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ∪ ∅ ) = ( 𝐴 × ( 𝐵 ∖ 𝐶 ) )
9	7 8	eqtri	⊢ ( ∅ ∪ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) ) = ( 𝐴 × ( 𝐵 ∖ 𝐶 ) )
10	1 6 9	3eqtrri	⊢ ( 𝐴 × ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 × 𝐵 ) ∖ ( 𝐴 × 𝐶 ) )