Theorem nel1nelini 45111

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

Hypothesis

Ref	Expression
nel1nelini.1	⊢ ¬ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
nel1nelini	⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

Proof of Theorem nel1nelini

Step	Hyp	Ref	Expression
1		nel1nelini.1	. 2 ⊢ ¬ 𝐴 ∈ 𝐵
2		nel1nelin 4178	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:  ¬ wn 3   ∈ wcel 2109   ∩ cin 3921

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-in 3929

This theorem is referenced by: (None)