Theorem elndif 4017

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 4016	. 2 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) → ¬ 𝐴 ∈ 𝐵)
2	1	con2i 141	1 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113   ∖ cdif 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-dif 3844

This theorem is referenced by:  peano5  7618  extmptsuppeq  7876  undifixp  8537  ssfin4  9803  isf32lem3  9848  isf34lem4  9870  xrinfmss  12779  restntr  21926  cmpcld  22146  reconnlem2  23572  lebnumlem1  23706  i1fd  24426  hgt750lemd  32190  fmlasucdisj  32924  dfon2lem6  33328  onsucconni  34256  meaiininclem  43550  caragendifcl  43578