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Theorem class2seteq 4192
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 2771 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 ax-1 6 . . . . 5  |-  ( A  e.  _V  ->  (
x  e.  A  ->  A  e.  _V )
)
32ralrimiv 2566 . . . 4  |-  ( A  e.  _V  ->  A. x  e.  A  A  e.  _V )
4 rabid2 2671 . . . 4  |-  ( A  =  { x  e.  A  |  A  e. 
_V }  <->  A. x  e.  A  A  e.  _V )
53, 4sylibr 134 . . 3  |-  ( A  e.  _V  ->  A  =  { x  e.  A  |  A  e.  _V } )
65eqcomd 2199 . 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 1364    e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-rab 2481  df-v 2762
This theorem is referenced by: (None)
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