Theorem elndif 4107

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 4106	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵)
2	1	con2i 141	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ∖ cdif 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-dif 3941

This theorem is referenced by:  peano5  7607  extmptsuppeq  7856  undifixp  8500  ssfin4  9734  isf32lem3  9779  isf34lem4  9801  xrinfmss  12706  restntr  21792  cmpcld  22012  reconnlem2  23437  lebnumlem1  23567  i1fd  24284  hgt750lemd  31921  fmlasucdisj  32648  dfon2lem6  33035  onsucconni  33787  meaiininclem  42775  caragendifcl  42803