Theorem canth 5796

Description: No set A is equinumerous to its power set (Cantor's theorem), i.e., no function can map A onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1488 if you want the form -. E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)

Hypothesis

Ref	Expression
canth.1	|- A e. _V

Assertion

Ref	Expression
canth	|- -. F : A -onto-> ~P A

Proof of Theorem canth

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

Step	Hyp	Ref	Expression
1		canth.1	. . . 4 |- A e. _V
2		ssrab2 3227	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
3	1, 2	elpwi2 4137	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
4		forn 5413	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
5	3, 4	eleqtrrid 2256	. 2 |- ( F : A -onto-> ~P A -> { x e. A | -. x e. ( F ` x ) } e. ran F )
6		pm5.19 696	. . . . . 6 |- -. ( y e. ( F ` y ) <-> -. y e. ( F ` y ) )
7		eleq2 2230	. . . . . . 7 |- ( ( F ` y ) = { x e. A | -. x e. ( F ` x ) } -> ( y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) )
8		id 19	. . . . . . . . . 10 |- ( x = y -> x = y )
9		fveq2 5486	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
10	8, 9	eleq12d 2237	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` x ) <-> y e. ( F ` y ) ) )
11	10	notbid 657	. . . . . . . 8 |- ( x = y -> ( -. x e. ( F ` x ) <-> -. y e. ( F ` y ) ) )
12	11	elrab3 2883	. . . . . . 7 |- ( y e. A -> ( y e. { x e. A | -. x e. ( F ` x ) } <-> -. y e. ( F ` y ) ) )
13	7, 12	sylan9bbr 459	. . . . . 6 |- ( ( y e. A /\ ( F ` y ) = { x e. A | -. x e. ( F ` x ) } ) -> ( y e. ( F ` y ) <-> -. y e. ( F ` y ) ) )
14	6, 13	mto 652	. . . . 5 |- -. ( y e. A /\ ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
15	14	imnani 681	. . . 4 |- ( y e. A -> -. ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
16	15	nrex 2558	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
17		fofn 5412	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
18		fvelrnb 5534	. . . 4 |- ( F Fn A -> ( { x e. A | -. x e. ( F ` x ) } e. ran F <-> E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) } ) )
19	17, 18	syl 14	. . 3 |- ( F : A -onto-> ~P A -> ( { x e. A | -. x e. ( F ` x ) } e. ran F <-> E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) } ) )
20	16, 19	mtbiri 665	. 2 |- ( F : A -onto-> ~P A -> -. { x e. A | -. x e. ( F ` x ) } e. ran F )
21	5, 20	pm2.65i 629	1 |- -. F : A -onto-> ~P A

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   {crab 2448   _Vcvv 2726   ~Pcpw 3559   ran crn 4605   Fn wfn 5183   -onto->wfo 5186   ` cfv 5188

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196

This theorem is referenced by: (None)