Theorem unipw 4218

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 3813	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3588	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1598	. . . 4 |- ( E. y ( x e. y /\ y e. ~P A ) -> x e. A )
4	1, 3	sylbi 121	. . 3 |- ( x e. U. ~P A -> x e. A )
5		vex 2741	. . . . 5 |- x e. _V
6	5	snid 3624	. . . 4 |- x e. { x }
7		snelpwi 4213	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3815	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 414	. . 3 |- ( x e. A -> x e. U. ~P A )
10	4, 9	impbii 126	. 2 |- ( x e. U. ~P A <-> x e. A )
11	10	eqriv 2174	1 |- U. ~P A = A

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   ~Pcpw 3576   {csn 3593   U.cuni 3810

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-uni 3811

This theorem is referenced by:  pwtr  4220  pwexb  4475  univ  4477  unixpss  4740  eltg4i  13558  distop  13588  distopon  13590  distps  13594  ntrss2  13624  isopn3  13628  discld  13639  txdis  13780