Theorem nelsn 3729

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 3712	. 2 |- ( A e. { B } -> A = B )
2	1	necon3ai 2463	1 |- ( A =/= B -> -. A e. { B } )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2205   =/= wne 2414   {csn 3694

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-sn 3700

This theorem is referenced by:  xnn0nnen  10823  nnoddn2prmb  12985