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Theorem rabsn 3710
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 2356 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21pm5.32ri 619 . . . 4  |-  ( ( x  e.  A  /\  x  =  B )  <->  ( B  e.  A  /\  x  =  B )
)
32baib 871 . . 3  |-  ( B  e.  A  ->  (
( x  e.  A  /\  x  =  B
)  <->  x  =  B
) )
43abbidv 2410 . 2  |-  ( B  e.  A  ->  { x  |  ( x  e.  A  /\  x  =  B ) }  =  { x  |  x  =  B } )
5 df-rab 2565 . 2  |-  { x  e.  A  |  x  =  B }  =  {
x  |  ( x  e.  A  /\  x  =  B ) }
6 df-sn 3659 . 2  |-  { B }  =  { x  |  x  =  B }
74, 5, 63eqtr4g 2353 1  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   {csn 3653
This theorem is referenced by:  unisn3  4539  sylow3lem6  14959  lineunray  24842  iint  25715  pmapat  30574  dia0  31864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-rab 2565  df-sn 3659
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