Theorem nelsn 4159

Description: If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)

Assertion

Ref	Expression
nelsn	⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Proof of Theorem nelsn

Step	Hyp	Ref	Expression
1		elsni 4142	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2	1	necon3ai 2807	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977   ≠ wne 2780  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-sn 4126

This theorem is referenced by:  fvunsn  6350  nnoddn2prmb  15356  lbsextlem4  18982  obslbs  19893  vdgr1b  26431  submateqlem1  29201  submateqlem2  29202  qqhval2  29354  derangsn  30406  clsk3nimkb  37358  clsk1indlem1  37363  disjf1o  38373  fperdvper  38808  dvnmul  38833  wallispi  38963  etransc  39176  gsumge0cl  39264  meadjiunlem  39358  hspmbllem2  39517  upgrres1  40532