Theorem neldif 4060

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

Assertion

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

Proof of Theorem neldif

Step	Hyp	Ref	Expression
1		eldif 3893	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
2	1	simplbi2 500	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ (𝐵 ∖ 𝐶)))
3	2	con1d 145	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐶))
4	3	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108   ∖ cdif 3880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886

This theorem is referenced by:  peano5  7714  peano5OLD  7715  boxcutc  8687  etransc  43714