Theorem pwuni 4178

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 3824	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2733	. . . 4 |- x e. _V
3	2	elpw 3572	. . 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 3151	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 2141   C_ wss 3121   ~Pcpw 3566   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797

This theorem is referenced by:  uniexb  4458  2pwuninelg  6262  istopon  12805  eltg3i  12850  mopnfss  13241