Theorem class2set 3471

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

Step	Hyp	Ref	Expression
1		rabexg 3460	. 2 |- (A e. _V -> {x e. A | A e. _V} e. _V)
2		simpl 346	. . . . 5 |- ((-. A e. _V /\ x e. A) -> -. A e. _V)
3	2	nrexdv 2193	. . . 4 |- (-. A e. _V -> -. E.x e. A A e. _V)
4		rabn0 2893	. . . . 5 |- ({x e. A | A e. _V} =/= (/) <-> E.x e. A A e. _V)
5	4	necon1bbii 2060	. . . 4 |- (-. E.x e. A A e. _V <-> {x e. A | A e. _V} = (/))
6	3, 5	sylib 215	. . 3 |- (-. A e. _V -> {x e. A | A e. _V} = (/))
7		0ex 3446	. . 3 |- (/) e. _V
8	6, 7	syl6eqel 1979	. 2 |- (-. A e. _V -> {x e. A | A e. _V} e. _V)
9	1, 8	pm2.61i 140	1 |- {x e. A | A e. _V} e. _V

Syntax hints:  -. wn 2   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292  (/)c0 2875

This theorem is referenced by:  abrexex 4836  fsum1s 8269  fsump1s 8273  fprod1s 14677  fprodp1s 14682

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876

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