Theorem ssnel 44031

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 3978	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵)
2	1	stoic1a 1772	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∈ wcel 2104   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3956  df-ss 3966

This theorem is referenced by:  nelrnres  44186  supminfxr2  44479