Theorem pwuni 4116

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 3764	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2689	. . . 4 |- x e. _V
3	2	elpw 3516	. . 3 |- ( x e. ~P U. A <-> x C_ U. A )
4	1, 3	sylibr 133	. 2 |- ( x e. A -> x e. ~P U. A )
5	4	ssriv 3101	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 1480   C_ wss 3071   ~Pcpw 3510   U.cuni 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737

This theorem is referenced by:  uniexb  4394  2pwuninelg  6180  istopon  12180  eltg3i  12225  mopnfss  12616