Theorem class2seteq 4082

Description: Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Assertion

Ref	Expression
class2seteq	|- ( A e. V -> { x e. A | A e. _V } = A )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elex 2692	. 2 |- ( A e. V -> A e. _V )
2		ax-1 6	. . . . 5 |- ( A e. _V -> ( x e. A -> A e. _V ) )
3	2	ralrimiv 2502	. . . 4 |- ( A e. _V -> A. x e. A A e. _V )
4		rabid2 2605	. . . 4 |- ( A = { x e. A | A e. _V } <-> A. x e. A A e. _V )
5	3, 4	sylibr 133	. . 3 |- ( A e. _V -> A = { x e. A | A e. _V } )
6	5	eqcomd 2143	. 2 |- ( A e. _V -> { x e. A | A e. _V } = A )
7	1, 6	syl 14	1 |- ( A e. V -> { x e. A | A e. _V } = A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   A.wral 2414   {crab 2418   _Vcvv 2681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-rab 2423  df-v 2683

This theorem is referenced by: (None)