Theorem ssdifin0

Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013) (Proof shortened by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Assertion	ssdifin0	⊢ ( 𝐴 ⊆ ( 𝐵 ∖ 𝐶 ) → ( 𝐴 ∩ 𝐶 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		ssrin	⊢ ( 𝐴 ⊆ ( 𝐵 ∖ 𝐶 ) → ( 𝐴 ∩ 𝐶 ) ⊆ ( ( 𝐵 ∖ 𝐶 ) ∩ 𝐶 ) )
2		incom	⊢ ( ( 𝐵 ∖ 𝐶 ) ∩ 𝐶 ) = ( 𝐶 ∩ ( 𝐵 ∖ 𝐶 ) )
3		disjdif	⊢ ( 𝐶 ∩ ( 𝐵 ∖ 𝐶 ) ) = ∅
4	2 3	eqtri	⊢ ( ( 𝐵 ∖ 𝐶 ) ∩ 𝐶 ) = ∅
5		sseq0	⊢ ( ( ( 𝐴 ∩ 𝐶 ) ⊆ ( ( 𝐵 ∖ 𝐶 ) ∩ 𝐶 ) ∧ ( ( 𝐵 ∖ 𝐶 ) ∩ 𝐶 ) = ∅ ) → ( 𝐴 ∩ 𝐶 ) = ∅ )
6	1 4 5	sylancl	⊢ ( 𝐴 ⊆ ( 𝐵 ∖ 𝐶 ) → ( 𝐴 ∩ 𝐶 ) = ∅ )