Theorem elnelall

Description: A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018)

	Ref	Expression
Assertion	elnelall	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∉ 𝐵 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
2		pm2.24	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ 𝐵 → 𝜑 ) )
3	1 2	syl5bi	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∉ 𝐵 → 𝜑 ) )