Theorem pwuni 4329

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 3963	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2824	. . . 4 |- x e. _V
3	2	elpw 3694	. . 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 3252	1 |- A C_ ~P U. A

This proof depends on syntax axioms:   e. wcel 2209   C_ wss 3220   ~Pcpw 3688   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-uni 3936

This theorem is used by:  uniexb  4619  2pwuninelg  6554  istopon  15114  eltg3i  15157  mopnfss  15548