Theorem unipw 3972

Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)

Assertion

Ref	Expression
unipw	|- U. ~P A = A

Proof of Theorem unipw

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3604	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3393	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1529	. . . 4 |- ( E. y ( x e. y /\ y e. ~P A ) -> x e. A )
4	1, 3	sylbi 119	. . 3 |- ( x e. U. ~P A -> x e. A )
5		vex 2604	. . . . 5 |- x e. _V
6	5	snid 3425	. . . 4 |- x e. { x }
7		snelpwi 3967	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3606	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 405	. . 3 |- ( x e. A -> x e. U. ~P A )
10	4, 9	impbii 124	. 2 |- ( x e. U. ~P A <-> x e. A )
11	10	eqriv 2078	1 |- U. ~P A = A

Syntax hints:   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   ~Pcpw 3382   {csn 3398   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-uni 3602

This theorem is referenced by:  pwtr  3974  pwexb  4224  univ  4225  unixpss  4469