Theorem unipw 3953

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 3583	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3370	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1489	. . . 4 |- ( E. y ( x e. y /\ y e. ~P A ) -> x e. A )
4	1, 3	sylbi 114	. . 3 |- ( x e. U. ~P A -> x e. A )
5		vex 2560	. . . . 5 |- x e. _V
6	5	snid 3402	. . . 4 |- x e. { x }
7		snelpwi 3948	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3585	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 393	. . 3 |- ( x e. A -> x e. U. ~P A )
10	4, 9	impbii 117	. 2 |- ( x e. U. ~P A <-> x e. A )
11	10	eqriv 2037	1 |- U. ~P A = A

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   ~Pcpw 3359   {csn 3375   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-uni 3581

This theorem is referenced by:  pwtr  3955  pwexb  4206  univ  4207  unixpss  4451