Theorem neldifsn 4722

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 2951	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifsni 4720	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴)
3	1, 2	mto 196	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Syntax hints:  ¬ wn 3   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-sn 4559

This theorem is referenced by:  neldifsnd  4723  fofinf1o  9024  dfac9  9823  xrsupss  12972  hashgt23el  14067  fvsetsid  16797  islbs3  20332  islindf4  20955  ufinffr  22988  i1fd  24750  finsumvtxdg2sstep  27819  matunitlindflem1  35700  poimirlem25  35729  itg2addnclem  35755  itg2addnclem2  35756  prter2  36822