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Theorem rabsn 4026
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 2520 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21pm5.32ri 638 . . . 4  |-  ( ( x  e.  A  /\  x  =  B )  <->  ( B  e.  A  /\  x  =  B )
)
32baib 896 . . 3  |-  ( B  e.  A  ->  (
( x  e.  A  /\  x  =  B
)  <->  x  =  B
) )
43abbidv 2584 . 2  |-  ( B  e.  A  ->  { x  |  ( x  e.  A  /\  x  =  B ) }  =  { x  |  x  =  B } )
5 df-rab 2801 . 2  |-  { x  e.  A  |  x  =  B }  =  {
x  |  ( x  e.  A  /\  x  =  B ) }
6 df-sn 3962 . 2  |-  { B }  =  { x  |  x  =  B }
74, 5, 63eqtr4g 2515 1  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   {cab 2435   {crab 2796   {csn 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-rab 2801  df-sn 3962
This theorem is referenced by:  unisn3  4192  sylow3lem6  16221  lineunray  28298  lco0  31054  pmapat  33689  dia0  34979
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