Theorem neldifsn 4748

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 2941	. 2 ⊢ ¬ 𝐴 ≠ 𝐴
2		eldifsni 4746	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐴}) → 𝐴 ≠ 𝐴)
3	1, 2	mto 197	1 ⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Syntax hints:  ¬ wn 3   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898  {csn 4580

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-sn 4581

This theorem is referenced by:  neldifsnd  4749  fofinf1o  9232  dfac9  10047  1div0  11796  xrsupss  13224  hashgt23el  14347  fvsetsid  17095  islbs3  21110  islindf4  21793  ufinffr  23873  i1fd  25638  finsumvtxdg2sstep  29623  matunitlindflem1  37817  poimirlem25  37846  itg2addnclem  37872  itg2addnclem2  37873  prter2  39151