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Theorem nelsn 3626
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
StepHypRef Expression
1 elsni 3609 . 2  |-  ( A  e.  { B }  ->  A  =  B )
21necon3ai 2396 1  |-  ( A  =/=  B  ->  -.  A  e.  { B } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2148    =/= wne 2347   {csn 3591
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-sn 3597
This theorem is referenced by:  nnoddn2prmb  12232
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