Theorem neldif 4088

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 3915	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
2	1	simplbi2 505	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ (𝐵 ∖ 𝐶)))
3	2	con1d 146	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐶))
4	3	imp 411	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2143   ∖ cdif 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908

This theorem is referenced by:  peano5  7886  boxcutc  8935  dflring3  33787  dflring4  33788  etransc  46997