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Theorem class2set 4396
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 4382 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
2 simpl 445 . . . . 5  |-  ( ( -.  A  e.  _V  /\  x  e.  A )  ->  -.  A  e.  _V )
32nrexdv 2815 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. x  e.  A  A  e.  _V )
4 rabn0 3632 . . . . 5  |-  ( { x  e.  A  |  A  e.  _V }  =/=  (/)  <->  E. x  e.  A  A  e.  _V )
54necon1bbii 2662 . . . 4  |-  ( -. 
E. x  e.  A  A  e.  _V  <->  { x  e.  A  |  A  e.  _V }  =  (/) )
63, 5sylib 190 . . 3  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  (/) )
7 0ex 4364 . . 3  |-  (/)  e.  _V
86, 7syl6eqel 2530 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
91, 8pm2.61i 159 1  |-  { x  e.  A  |  A  e.  _V }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1727   E.wrex 2712   {crab 2715   _Vcvv 2962   (/)c0 3613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-in 3313  df-ss 3320  df-nul 3614
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