Theorem neldif 4109

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 3949	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
2	1	simplbi2 503	. . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ (𝐵 ∖ 𝐶)))
3	2	con1d 147	. 2 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐶))
4	3	imp 409	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∈ wcel 2113   ∖ cdif 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-dif 3942

This theorem is referenced by:  peano5  7608  boxcutc  8508  etransc  42575