Theorem pwuni 4288

Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)

Assertion

Ref	Expression
pwuni	|- A C_ ~P U. A

Proof of Theorem pwuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elssuni 3926	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2806	. . . 4 |- x e. _V
3	2	elpw 3662	. . 3 |- ( x e. ~P U. A <-> x C_ U. A )
4	1, 3	sylibr 134	. 2 |- ( x e. A -> x e. ~P U. A )
5	4	ssriv 3232	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-uni 3899

This theorem is referenced by:  uniexb  4576  2pwuninelg  6492  istopon  14807  eltg3i  14850  mopnfss  15241