Theorem neldifsnd 4747

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 4746	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ∖ cdif 3902  {csn 4579

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3440  df-dif 3908  df-sn 4580

This theorem is referenced by:  difsnb  4760  fsnunf2  7126  fsumsplit1  15670  rpnnen2lem9  16149  fprodfvdvdsd  16263  ramub1lem1  16956  ramub1lem2  16957  prmdvdsprmo  16972  acsfiindd  18477  gsummgp0  20221  islindf4  21763  gsummatr01lem3  22560  nbgrnself  29322  omsmeas  34290  onint1  36422  bj-fvsnun2  37229  poimirlem30  37629  prtlem80  38839  aks6d1c5lem3  42110  gneispace0nelrn3  44115  supminfxr2  45449  fsumnncl  45554  hoidmv1lelem2  46574  hspmbllem1  46608  hspmbllem2  46609  fsumsplitsndif  47358  isubgr3stgrlem3  47953  mgpsumunsn  48346