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Metamath Proof Explorer

Theorem sscond

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

		Ref	Expression
	Hypothesis	ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	Assertion	sscond	⊢ ( 𝜑 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssdifd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sscon	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) )