MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabsn Structured version   Unicode version

Theorem rabsn 4083
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 2526 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21pm5.32ri 636 . . . 4  |-  ( ( x  e.  A  /\  x  =  B )  <->  ( B  e.  A  /\  x  =  B )
)
32baib 901 . . 3  |-  ( B  e.  A  ->  (
( x  e.  A  /\  x  =  B
)  <->  x  =  B
) )
43abbidv 2590 . 2  |-  ( B  e.  A  ->  { x  |  ( x  e.  A  /\  x  =  B ) }  =  { x  |  x  =  B } )
5 df-rab 2813 . 2  |-  { x  e.  A  |  x  =  B }  =  {
x  |  ( x  e.  A  /\  x  =  B ) }
6 df-sn 4017 . 2  |-  { B }  =  { x  |  x  =  B }
74, 5, 63eqtr4g 2520 1  |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-rab 2813  df-sn 4017
This theorem is referenced by:  unisn3  4252  sylow3lem6  16854  lineunray  30028  nzss  31466  lco0  33301  pmapat  35903  dia0  37195
  Copyright terms: Public domain W3C validator