Theorem elndif 4072

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 4071	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵)
2	1	con2i 140	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ∖ cdif 4048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050

This theorem is used by:  peano5  7939  extmptsuppeq  8243  undifixp  9000  ssfin4  10427  isf32lem3  10472  isf34lem4  10494  xrinfmss  13480  ssdifidlprm  21693  restntr  23555  cmpcld  23775  reconnlem2  25202  lebnumlem1  25337  i1fd  26057  plngrotlem1  29322  plngrotlem2  29323  dflringlem3  34132  dflring3  34133  dflring4  34134  hgt750lemd  35388  fmlasucdisj  36372  dfon2lem6  36765  onsucconni  37440  meaiininclem  47769  caragendifcl  47797