Theorem ssnel 45864

Description: If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
ssnel	⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)

Proof of Theorem ssnel

Step	Hyp	Ref	Expression
1		ssel2 3929	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
2	1	stoic1a 1805	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2837  df-ss 3919

This theorem is used by:  nelrnres  46006  supminfxr2  46284