Theorem neldifsnd 4686

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 4685	. 2 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
2	1	a1i 11	1 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   ∖ cdif 3878  {csn 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-sn 4526

This theorem is referenced by:  difsnb  4699  fsnunf2  6925  rpnnen2lem9  15567  fprodfvdvdsd  15675  ramub1lem1  16352  ramub1lem2  16353  prmdvdsprmo  16368  acsfiindd  17779  gsummgp0  19354  islindf4  20527  gsummatr01lem3  21262  nbgrnself  27149  omsmeas  31691  onint1  33910  bj-fvsnun2  34671  poimirlem30  35087  prtlem80  36157  gneispace0nelrn3  40845  supminfxr2  42108  fsumnncl  42213  fsumsplit1  42214  hoidmv1lelem2  43231  hspmbllem1  43265  hspmbllem2  43266  fsumsplitsndif  43890  mgpsumunsn  44763