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Theorem class2set 4180
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 4166 . 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 2648 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
E. x  e.  A  A  e.  _V )
4 rabn0 3476 . . . . 5  |-  ( { x  e.  A  |  A  e.  _V }  =/=  (/)  <->  E. x  e.  A  A  e.  _V )
54necon1bbii 2500 . . . 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 4152 . . 3  |-  (/)  e.  _V
86, 7syl6eqel 2373 . 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 1625    e. wcel 1686   E.wrex 2546   {crab 2549   _Vcvv 2790   (/)c0 3457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-in 3161  df-ss 3168  df-nul 3458
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