Theorem nelrdva

Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017)

	Ref	Expression
Hypothesis	nelrdva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≠ 𝐵 )
Assertion	nelrdva	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nelrdva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≠ 𝐵 )
2		eqidd	⊢ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 = 𝐵 )
3		eleq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) )
4	3	anbi2d	⊢ ( 𝑥 = 𝐵 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) ) )
5		neeq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 ≠ 𝐵 ↔ 𝐵 ≠ 𝐵 ) )
6	4 5	imbi12d	⊢ ( 𝑥 = 𝐵 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≠ 𝐵 ) ↔ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≠ 𝐵 ) ) )
7	6 1	vtoclg	⊢ ( 𝐵 ∈ 𝐴 → ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≠ 𝐵 ) )
8	7	anabsi7	⊢ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≠ 𝐵 )
9	8	neneqd	⊢ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → ¬ 𝐵 = 𝐵 )
10	2 9	pm2.65da	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐴 )