Theorem elndif 4143

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 4142	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵)
2	1	con2i 139	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   ∖ cdif 3960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966

This theorem is referenced by:  peano5  7916  extmptsuppeq  8212  undifixp  8973  ssfin4  10348  isf32lem3  10393  isf34lem4  10415  xrinfmss  13349  restntr  23206  cmpcld  23426  reconnlem2  24863  lebnumlem1  25007  i1fd  25730  ssdifidlprm  33466  hgt750lemd  34642  fmlasucdisj  35384  dfon2lem6  35770  onsucconni  36420  meaiininclem  46442  caragendifcl  46470