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Theorem class2seteq 3939
Description: Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
Assertion
Ref Expression
class2seteq  |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem class2seteq
StepHypRef Expression
1 elex 2611 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 ax-1 5 . . . . 5  |-  ( A  e.  _V  ->  (
x  e.  A  ->  A  e.  _V )
)
32ralrimiv 2434 . . . 4  |-  ( A  e.  _V  ->  A. x  e.  A  A  e.  _V )
4 rabid2 2531 . . . 4  |-  ( A  =  { x  e.  A  |  A  e. 
_V }  <->  A. x  e.  A  A  e.  _V )
53, 4sylibr 132 . . 3  |-  ( A  e.  _V  ->  A  =  { x  e.  A  |  A  e.  _V } )
65eqcomd 2087 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  A )
71, 6syl 14 1  |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   A.wral 2349   {crab 2353   _Vcvv 2602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-rab 2358  df-v 2604
This theorem is referenced by: (None)
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