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Theorem rabsn 4010
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 2494 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21pm5.32ri 642 . . . 4  |-  ( ( x  e.  A  /\  x  =  B )  <->  ( B  e.  A  /\  x  =  B )
)
32baib 911 . . 3  |-  ( B  e.  A  ->  (
( x  e.  A  /\  x  =  B
)  <->  x  =  B
) )
43abbidv 2546 . 2  |-  ( B  e.  A  ->  { x  |  ( x  e.  A  /\  x  =  B ) }  =  { x  |  x  =  B } )
5 df-rab 2723 . 2  |-  { x  e.  A  |  x  =  B }  =  {
x  |  ( x  e.  A  /\  x  =  B ) }
6 df-sn 3942 . 2  |-  { B }  =  { x  |  x  =  B }
74, 5, 63eqtr4g 2487 1  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2414   {crab 2718   {csn 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-rab 2723  df-sn 3942
This theorem is referenced by:  unisn3  4179  sylow3lem6  17227  lineunray  30863  pmapat  33240  dia0  34532  nzss  36579  lco0  39823
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