Theorem ssdif

Description: Difference law for subsets. (Contributed by NM, 28-May-1998)

	Ref	Expression
Assertion	ssdif	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	anim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶 ) → ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
3		eldif	⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶 ) )
4		eldif	⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) )
5	2 3 4	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) → 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) )
6	5	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∖ 𝐶 ) ⊆ ( 𝐵 ∖ 𝐶 ) )