Theorem neldif 3697

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 3550	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
2	1	simplbi2 653	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ (𝐵 ∖ 𝐶)))
3	2	con1d 138	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐶))
4	3	imp 444	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543

This theorem is referenced by:  peano5  6981  boxcutc  7837  etransc  39176