Theorem neldifsn 4794

Description: The class 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
neldifsn	⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Proof of Theorem neldifsn

Step	Hyp	Ref	Expression
1		neirr 2947	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifsni 4792	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴)
3	1, 2	mto 196	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Syntax hints:  ¬ wn 3   ∈ wcel 2104   ≠ wne 2938   ∖ cdif 3944  {csn 4627

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-v 3474  df-dif 3950  df-sn 4628

This theorem is referenced by:  neldifsnd  4795  fofinf1o  9329  dfac9  10133  xrsupss  13292  hashgt23el  14388  fvsetsid  17105  islbs3  20913  islindf4  21612  ufinffr  23653  i1fd  25430  finsumvtxdg2sstep  29073  matunitlindflem1  36787  poimirlem25  36816  itg2addnclem  36842  itg2addnclem2  36843  prter2  38054