Theorem neldifsnd 4726

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 4725	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ∖ cdif 3933  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3939  df-sn 4568

This theorem is referenced by:  difsnb  4739  fsnunf2  6948  rpnnen2lem9  15575  fprodfvdvdsd  15683  ramub1lem1  16362  ramub1lem2  16363  prmdvdsprmo  16378  acsfiindd  17787  gsummgp0  19358  islindf4  20982  gsummatr01lem3  21266  nbgrnself  27141  omsmeas  31581  onint1  33797  bj-fvsnun2  34541  poimirlem30  34937  prtlem80  36012  gneispace0nelrn3  40512  supminfxr2  41765  fsumnncl  41872  fsumsplit1  41873  hoidmv1lelem2  42894  hspmbllem1  42928  hspmbllem2  42929  fsumsplitsndif  43553  mgpsumunsn  44429