Theorem class2seteq 4211

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 2784	. 2 |- ( A e. V -> A e. _V )
2		ax-1 6	. . . . 5 |- ( A e. _V -> ( x e. A -> A e. _V ) )
3	2	ralrimiv 2579	. . . 4 |- ( A e. _V -> A. x e. A A e. _V )
4		rabid2 2684	. . . 4 |- ( A = { x e. A | A e. _V } <-> A. x e. A A e. _V )
5	3, 4	sylibr 134	. . 3 |- ( A e. _V -> A = { x e. A | A e. _V } )
6	5	eqcomd 2212	. 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 1373   e. wcel 2177   A.wral 2485   {crab 2489   _Vcvv 2773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-rab 2494  df-v 2775

This theorem is referenced by: (None)