Theorem unipw 4008

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 3630	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3417	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1530	. . . 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 2615	. . . . 5 |- x e. _V
6	5	snid 3449	. . . 4 |- x e. { x }
7		snelpwi 4003	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3632	. . . 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 2080	1 |- U. ~P A = A

Syntax hints:   /\ wa 102   = wceq 1285   E.wex 1422   e. wcel 1434   ~Pcpw 3406   {csn 3422   U.cuni 3627

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-uni 3628

This theorem is referenced by:  pwtr  4010  pwexb  4260  univ  4261  unixpss  4509