Theorem elndif 4073

Description: A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

Assertion

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

Proof of Theorem elndif

Step	Hyp	Ref	Expression
1		eldifn 4072	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵)
2	1	con2i 139	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ∖ cdif 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892

This theorem is referenced by:  peano5  7844  extmptsuppeq  8138  undifixp  8882  ssfin4  10232  isf32lem3  10277  isf34lem4  10299  xrinfmss  13262  restntr  23147  cmpcld  23367  reconnlem2  24793  lebnumlem1  24928  i1fd  25648  ssdifidlprm  33518  hgt750lemd  34792  fmlasucdisj  35581  dfon2lem6  35968  onsucconni  36619  meaiininclem  46914  caragendifcl  46942