Theorem nelsn 3668

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 3651	. 2 |- ( A e. { B } -> A = B )
2	1	necon3ai 2425	1 |- ( A =/= B -> -. A e. { B } )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2176   =/= wne 2376   {csn 3633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-sn 3639

This theorem is referenced by:  xnn0nnen  10582  nnoddn2prmb  12585