Theorem neldifsnd 4756

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 4755	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2145   ∖ cdif 3904  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-sn 4586

This theorem is referenced by:  difsnb  4769  fsnunf2  7174  fsumsplit1  15786  rpnnen2lem9  16268  fprodfvdvdsd  16382  ramub1lem1  17076  ramub1lem2  17077  prmdvdsprmo  17092  acsfiindd  18599  gsummgp0  20390  islindf4  21948  gsummatr01lem3  22775  nbgrnself  29618  evlextv  33849  esplyindfv  33883  vietalem  33886  omsmeas  34630  onint1  36822  bj-fvsnun2  37760  poimirlem30  38161  prtlem80  39497  aks6d1c5lem3  42766  gneispace0nelrn3  44730  supminfxr2  46041  fsumnncl  46146  hoidmv1lelem2  47164  hspmbllem1  47198  hspmbllem2  47199  fsumsplitsndif  47973  isubgr3stgrlem3  48588  mgpsumunsn  48992