Theorem ssdif2d

Description: If A is contained in B and C is contained in D , then ( A \ D ) is contained in ( B \ C ) . Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	ssdif2d.2	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )
Assertion	ssdif2d	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐷 ) ⊆ ( 𝐵 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssdif2d.2	⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )
3	2	sscond	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐷 ) ⊆ ( 𝐴 ∖ 𝐶 ) )
4	1	ssdifd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )
5	3 4	sstrd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐷 ) ⊆ ( 𝐵 ∖ 𝐶 ) )