Theorem unipwr 39068

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4918. The proof of this theorem was automatically generated from unipwrVD 39067 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem unipwr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- x e. _V
2	1	snid 4208	. . 3 |- x e. { x }
3		snelpwi 4912	. . 3 |- ( x e. A -> { x } e. ~P A )
4		elunii 4441	. . 3 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
5	2, 3, 4	sylancr 695	. 2 |- ( x e. A -> x e. U. ~P A )
6	5	ssriv 3607	1 |- A C_ U. ~P A

Syntax hints:   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)