Theorem nel2nelin 4108

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 4102	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶)
2	1	con3i 155	1 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ∩ cin 4092

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094

This theorem is used by:  nel2nelini  46366  tposres  50200