Theorem neldif

Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994)

	Ref	Expression
Assertion	neldif	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ) → 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eldif	⊢ ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) )
2	1	simplbi2	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ) )
3	2	con1d	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
4	3	imp	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ) → 𝐴 ∈ 𝐶 )