Theorem pm2.24nel 3079

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

Assertion

Ref	Expression
pm2.24nel	⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))

Proof of Theorem pm2.24nel

Step	Hyp	Ref	Expression
1		df-nel 3067	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		pm2.24 125	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 → 𝜑))
3	1, 2	biimtrid 245	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146   ∉ wnel 3066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-nel 3067

This theorem is used by:  xnn0lenn0nn0  13283  ge2nprmge4  16778  afv2orxorb  47998  ppivalnnnprm  48413  isubgr3stgrlem6  48769