Theorem nel1nelin 39857

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

Assertion

Ref	Expression
nel1nelin	⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))

Proof of Theorem nel1nelin

Step	Hyp	Ref	Expression
1		elinel1 3950	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)
2	1	con3i 151	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2145   ∩ cin 3722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730

This theorem is referenced by:  nel1nelini  39860