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Theorem class2set 2724
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.
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 2714 . 2 |- (A e. V -> {x e. A | A e. V} e. V)
2 pm3.26 319 . . . . 5 |- ((-. A e. V /\ x e. A) -> -. A e. V)
32nrexdv 1722 . . . 4 |- (-. A e. V -> -. E.x e. A A e. V)
4 rabn0 2282 . . . . 5 |- ({x e. A | A e. V} =/= (/) <-> E.x e. A A e. V)
54necon1bbii 1609 . . . 4 |- (-. E.x e. A A e. V <-> {x e. A | A e. V} = (/))
63, 5sylib 198 . . 3 |- (-. A e. V -> {x e. A | A e. V} = (/))
7 0ex 2701 . . 3 |- (/) e. V
86, 7syl6eqel 1548 . 2 |- (-. A e. V -> {x e. A | A e. V} e. V)
91, 8pm2.61i 126 1 |- {x e. A | A e. V} e. V
Colors of variables: wff set class
Syntax hints:  -. wn 2   = wceq 953   e. wcel 955  E.wrex 1638  {crab 1640  Vcvv 1802  (/)c0 2270
This theorem is referenced by:  abrexex 3845  fsum1s 6947  fsump1s 6951
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043  df-nul 2271
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