Theorem nelsn 3610

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 3593	. 2 |- ( A e. { B } -> A = B )
2	1	necon3ai 2384	1 |- ( A =/= B -> -. A e. { B } )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2136   =/= wne 2335   {csn 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-v 2727  df-sn 3581

This theorem is referenced by:  nnoddn2prmb  12190