Theorem nel1nelini 46267

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 4103	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2144   ∩ cin 4088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4090

This theorem is used by: (None)