Theorem pwnss 4830

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
pwnss	|- ( A e. V -> -. ~P A C_ A )

Proof of Theorem pwnss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq12 2691	. . . . . . 7 |- ( ( y = { x e. A | x e/ x } /\ y = { x e. A | x e/ x } ) -> ( y e. y <-> { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
2	1	anidms 677	. . . . . 6 |- ( y = { x e. A | x e/ x } -> ( y e. y <-> { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
3	2	notbid 308	. . . . 5 |- ( y = { x e. A | x e/ x } -> ( -. y e. y <-> -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
4		df-nel 2898	. . . . . . 7 |- ( x e/ x <-> -. x e. x )
5		eleq12 2691	. . . . . . . . 9 |- ( ( x = y /\ x = y ) -> ( x e. x <-> y e. y ) )
6	5	anidms 677	. . . . . . . 8 |- ( x = y -> ( x e. x <-> y e. y ) )
7	6	notbid 308	. . . . . . 7 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
8	4, 7	syl5bb 272	. . . . . 6 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
9	8	cbvrabv 3199	. . . . 5 |- { x e. A | x e/ x } = { y e. A | -. y e. y }
10	3, 9	elrab2 3366	. . . 4 |- ( { x e. A | x e/ x } e. { x e. A | x e/ x } <-> ( { x e. A | x e/ x } e. A /\ -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) )
11		pclem6 971	. . . 4 |- ( ( { x e. A | x e/ x } e. { x e. A | x e/ x } <-> ( { x e. A | x e/ x } e. A /\ -. { x e. A | x e/ x } e. { x e. A | x e/ x } ) ) -> -. { x e. A | x e/ x } e. A )
12	10, 11	ax-mp 5	. . 3 |- -. { x e. A | x e/ x } e. A
13		ssel 3597	. . 3 |- ( ~P A C_ A -> ( { x e. A | x e/ x } e. ~P A -> { x e. A | x e/ x } e. A ) )
14	12, 13	mtoi 190	. 2 |- ( ~P A C_ A -> -. { x e. A | x e/ x } e. ~P A )
15		ssrab2 3687	. . 3 |- { x e. A | x e/ x } C_ A
16		elpw2g 4827	. . 3 |- ( A e. V -> ( { x e. A | x e/ x } e. ~P A <-> { x e. A | x e/ x } C_ A ) )
17	15, 16	mpbiri 248	. 2 |- ( A e. V -> { x e. A | x e/ x } e. ~P A )
18	14, 17	nsyl3 133	1 |- ( A e. V -> -. ~P A C_ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   C_ wss 3574   ~Pcpw 4158

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nel 2898  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  pwne  4831  pwuninel2  7400