Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  class2set Unicode version

Theorem class2set 4072
 Description: Construct, from any class , 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
Distinct variable group:   ,

Proof of Theorem class2set
StepHypRef Expression
1 rabexg 4060 . 2
2 simpl 445 . . . . 5
32nrexdv 2608 . . . 4
4 rabn0 3381 . . . . 5
54necon1bbii 2464 . . . 4
63, 5sylib 190 . . 3
7 0ex 4047 . . 3
86, 7syl6eqel 2341 . 2
91, 8pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 5   wceq 1619   wcel 1621  wrex 2510  crab 2512  cvv 2727  c0 3362 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-in 3085  df-ss 3089  df-nul 3363
 Copyright terms: Public domain W3C validator