Theorem pwnss 4207

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 2271	. . . . . . 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 669	. . . . 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 2473	. . . . . . 7 |- ( x e/ x <-> -. x e. x )
5		eleq12 2271	. . . . . . . . 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 669	. . . . . . 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 2772	. . . . 5 |- { x e. A | x e/ x } = { y e. A | -. y e. y }
10	3, 9	elrab2 2933	. . . 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 1394	. . . 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 3188	. . 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 666	. 2 |- ( ~P A C_ A -> -. { x e. A | x e/ x } e. ~P A )
15		ssrab2 3279	. . 3 |- { x e. A | x e/ x } C_ A
16		elpw2g 4204	. . 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 627	1 |- ( A e. V -> -. ~P A C_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   e/ wnel 2472   {crab 2489   C_ wss 3167   ~Pcpw 3617

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4166

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-nel 2473  df-rab 2494  df-v 2775  df-in 3173  df-ss 3180  df-pw 3619

This theorem is referenced by:  pwne  4208  pwuninel2  6375