Theorem nelsn 4212

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 e. { B } )

Proof of Theorem nelsn

Step	Hyp	Ref	Expression
1		elsni 4194	. 2 |- ( A e. { B } -> A = B )
2	1	necon3ai 2819	1 |- ( A =/= B -> -. A e. { B } )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   =/= wne 2794   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-sn 4178

This theorem is referenced by:  fvunsn  6445  nnoddn2prmb  15518  lbsextlem4  19161  cnfldfunALT  19759  obslbs  20074  upgrres1  26205  submateqlem1  29873  submateqlem2  29874  qqhval2  30026  derangsn  31152  clsk3nimkb  38338  clsk1indlem1  38343  disjf1o  39378  cnrefiisplem  40055  fperdvper  40133  dvnmul  40158  wallispi  40287  etransc  40500  gsumge0cl  40588  meadjiunlem  40682  hspmbllem2  40841