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Theorem class2set 4327
Description: Construct, from any class  A, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)
Assertion
Ref Expression
class2set  |-  { x  e.  A  |  A  e.  _V }  e.  _V
Distinct variable group:    x, A

Proof of Theorem class2set
StepHypRef Expression
1 rabexg 4313 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
2 simpl 444 . . . . 5  |-  ( ( -.  A  e.  _V  /\  x  e.  A )  ->  -.  A  e.  _V )
32nrexdv 2769 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. x  e.  A  A  e.  _V )
4 rabn0 3607 . . . . 5  |-  ( { x  e.  A  |  A  e.  _V }  =/=  (/)  <->  E. x  e.  A  A  e.  _V )
54necon1bbii 2619 . . . 4  |-  ( -. 
E. x  e.  A  A  e.  _V  <->  { x  e.  A  |  A  e.  _V }  =  (/) )
63, 5sylib 189 . . 3  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  (/) )
7 0ex 4299 . . 3  |-  (/)  e.  _V
86, 7syl6eqel 2492 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
91, 8pm2.61i 158 1  |-  { x  e.  A  |  A  e.  _V }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916   (/)c0 3588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589
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