Theorem pwnss 4138

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 2231	. . . . . . 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 395	. . . . . 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 657	. . . . 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 2432	. . . . . . 7 |- ( x e/ x <-> -. x e. x )
5		eleq12 2231	. . . . . . . . 9 |- ( ( x = y /\ x = y ) -> ( x e. x <-> y e. y ) )
6	5	anidms 395	. . . . . . . 8 |- ( x = y -> ( x e. x <-> y e. y ) )
7	6	notbid 657	. . . . . . 7 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
8	4, 7	syl5bb 191	. . . . . 6 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
9	8	cbvrabv 2725	. . . . 5 |- { x e. A | x e/ x } = { y e. A | -. y e. y }
10	3, 9	elrab2 2885	. . . 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 1364	. . . 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 3136	. . 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 654	. 2 |- ( ~P A C_ A -> -. { x e. A | x e/ x } e. ~P A )
15		ssrab2 3227	. . 3 |- { x e. A | x e/ x } C_ A
16		elpw2g 4135	. . 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 167	. 2 |- ( A e. V -> { x e. A | x e/ x } e. ~P A )
18	14, 17	nsyl3 616	1 |- ( A e. V -> -. ~P A C_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   e/ wnel 2431   {crab 2448   C_ wss 3116   ~Pcpw 3559

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-nel 2432  df-rab 2453  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  pwne  4139  pwuninel2  6250