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Metamath Proof Explorer

Theorem ssneldd

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

	Ref	Expression
Hypotheses	ssneld.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	ssneldd.2	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐵 )
Assertion	ssneldd	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ssneld.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssneldd.2	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐵 )
3	1	ssneld	⊢ ( 𝜑 → ( ¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴 ) )
4	2 3	mpd	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )