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Metamath Proof Explorer

Theorem ssneld

Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017)

		Ref	Expression
	Hypothesis	ssneld.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	Assertion	ssneld	⊢ ( 𝜑 → ( ¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ssneld.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2	1	sseld	⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵 ) )
3	2	con3d	⊢ ( 𝜑 → ( ¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴 ) )