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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