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Theorem rabsn 3560
Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
Assertion
Ref Expression
rabsn  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Distinct variable groups:    x, A    x, B

Proof of Theorem rabsn
StepHypRef Expression
1 eleq1 2180 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21pm5.32ri 450 . . . 4  |-  ( ( x  e.  A  /\  x  =  B )  <->  ( B  e.  A  /\  x  =  B )
)
32baib 889 . . 3  |-  ( B  e.  A  ->  (
( x  e.  A  /\  x  =  B
)  <->  x  =  B
) )
43abbidv 2235 . 2  |-  ( B  e.  A  ->  { x  |  ( x  e.  A  /\  x  =  B ) }  =  { x  |  x  =  B } )
5 df-rab 2402 . 2  |-  { x  e.  A  |  x  =  B }  =  {
x  |  ( x  e.  A  /\  x  =  B ) }
6 df-sn 3503 . 2  |-  { B }  =  { x  |  x  =  B }
74, 5, 63eqtr4g 2175 1  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397   {csn 3497
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-rab 2402  df-sn 3503
This theorem is referenced by:  unisn3  4336
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