Theorem neldifsnd 4760

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 4759	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ∖ cdif 3914  {csn 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-sn 4593

This theorem is referenced by:  difsnb  4773  fsnunf2  7163  fsumsplit1  15718  rpnnen2lem9  16197  fprodfvdvdsd  16311  ramub1lem1  17004  ramub1lem2  17005  prmdvdsprmo  17020  acsfiindd  18519  gsummgp0  20234  islindf4  21754  gsummatr01lem3  22551  nbgrnself  29293  omsmeas  34321  onint1  36444  bj-fvsnun2  37251  poimirlem30  37651  prtlem80  38861  aks6d1c5lem3  42132  gneispace0nelrn3  44138  supminfxr2  45472  fsumnncl  45577  hoidmv1lelem2  46597  hspmbllem1  46631  hspmbllem2  46632  fsumsplitsndif  47378  isubgr3stgrlem3  47971  mgpsumunsn  48353