Theorem nel2nelin 4188

Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Assertion

Ref	Expression
nel2nelin	⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))

Proof of Theorem nel2nelin

Step	Hyp	Ref	Expression
1		elinel2 4182	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶)
2	1	con3i 154	1 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2107   ∩ cin 3930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-in 3938

This theorem is referenced by:  nel2nelini  45084  tposres  48717