Theorem pwnss 4161

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 2242	. . . . . . 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 397	. . . . . 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 667	. . . . 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 2443	. . . . . . 7 |- ( x e/ x <-> -. x e. x )
5		eleq12 2242	. . . . . . . . 9 |- ( ( x = y /\ x = y ) -> ( x e. x <-> y e. y ) )
6	5	anidms 397	. . . . . . . 8 |- ( x = y -> ( x e. x <-> y e. y ) )
7	6	notbid 667	. . . . . . 7 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
8	4, 7	bitrid 192	. . . . . 6 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
9	8	cbvrabv 2738	. . . . 5 |- { x e. A | x e/ x } = { y e. A | -. y e. y }
10	3, 9	elrab2 2898	. . . 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 1374	. . . 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 3151	. . 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 664	. 2 |- ( ~P A C_ A -> -. { x e. A | x e/ x } e. ~P A )
15		ssrab2 3242	. . 3 |- { x e. A | x e/ x } C_ A
16		elpw2g 4158	. . 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 168	. 2 |- ( A e. V -> { x e. A | x e/ x } e. ~P A )
18	14, 17	nsyl3 626	1 |- ( A e. V -> -. ~P A C_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   e/ wnel 2442   {crab 2459   C_ wss 3131   ~Pcpw 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-nel 2443  df-rab 2464  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by:  pwne  4162  pwuninel2  6285