Theorem elnelall 2527

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

Assertion

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

Proof of Theorem elnelall

Step	Hyp	Ref	Expression
1		df-nel 2516	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		pm2.24 630	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝜑))
3	1, 2	biimtrid 152	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2209   ∉ wnel 2515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in2 624

This proof depends on definitions:  df-bi 117  df-nel 2516

This theorem is used by:  xnn0lenn0nn0  10278