Theorem neldifsnd 4512

Description: The class 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
neldifsnd	⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Proof of Theorem neldifsnd

Step	Hyp	Ref	Expression
1		neldifsn 4511	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2157   ∖ cdif 3766  {csn 4368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-sn 4369

This theorem is referenced by:  difsnb  4525  fsnunf2  6681  rpnnen2lem9  15287  fprodfvdvdsd  15394  ramub1lem1  16063  ramub1lem2  16064  prmdvdsprmo  16079  acsfiindd  17492  gsummgp0  18924  islindf4  20502  gsummatr01lem3  20790  nbgrnself  26597  omsmeas  30901  onint1  32956  poimirlem30  33928  prtlem80  34882  gneispace0nelrn3  39222  supminfxr2  40442  fsumnncl  40547  fsumsplit1  40548  hoidmv1lelem2  41552  hspmbllem1  41586  hspmbllem2  41587  fsumsplitsndif  42183  mgpsumunsn  42939