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Metamath Proof Explorer

Theorem ssdifss

Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007)

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

Step	Hyp	Ref	Expression
1		difss	⊢ ( 𝐴 ∖ 𝐶 ) ⊆ 𝐴
2		sstr	⊢ ( ( ( 𝐴 ∖ 𝐶 ) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∖ 𝐶 ) ⊆ 𝐵 )
3	1 2	mpan	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∖ 𝐶 ) ⊆ 𝐵 )