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Theorem class2set 4178
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 4164 . 2  |-  ( A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
2 simpl 443 . . . . 5  |-  ( ( -.  A  e.  _V  /\  x  e.  A )  ->  -.  A  e.  _V )
32nrexdv 2646 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. x  e.  A  A  e.  _V )
4 rabn0 3474 . . . . 5  |-  ( { x  e.  A  |  A  e.  _V }  =/=  (/)  <->  E. x  e.  A  A  e.  _V )
54necon1bbii 2498 . . . 4  |-  ( -. 
E. x  e.  A  A  e.  _V  <->  { x  e.  A  |  A  e.  _V }  =  (/) )
63, 5sylib 188 . . 3  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  =  (/) )
7 0ex 4150 . . 3  |-  (/)  e.  _V
86, 7syl6eqel 2371 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  A  |  A  e.  _V }  e.  _V )
91, 8pm2.61i 156 1  |-  { x  e.  A  |  A  e.  _V }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   (/)c0 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456
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