Theorem canth 5875

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 1514 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 3268	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
3	1, 2	elpwi2 4191	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
4		forn 5483	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
5	3, 4	eleqtrrid 2286	. 2 |- ( F : A -onto-> ~P A -> { x e. A | -. x e. ( F ` x ) } e. ran F )
6		pm5.19 707	. . . . . 6 |- -. ( y e. ( F ` y ) <-> -. y e. ( F ` y ) )
7		eleq2 2260	. . . . . . 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 5558	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
10	8, 9	eleq12d 2267	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` x ) <-> y e. ( F ` y ) ) )
11	10	notbid 668	. . . . . . . 8 |- ( x = y -> ( -. x e. ( F ` x ) <-> -. y e. ( F ` y ) ) )
12	11	elrab3 2921	. . . . . . 7 |- ( y e. A -> ( y e. { x e. A | -. x e. ( F ` x ) } <-> -. y e. ( F ` y ) ) )
13	7, 12	sylan9bbr 463	. . . . . 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 663	. . . . 5 |- -. ( y e. A /\ ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
15	14	imnani 692	. . . 4 |- ( y e. A -> -. ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
16	15	nrex 2589	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
17		fofn 5482	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
18		fvelrnb 5608	. . . 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 676	. 2 |- ( F : A -onto-> ~P A -> -. { x e. A | -. x e. ( F ` x ) } e. ran F )
21	5, 20	pm2.65i 640	1 |- -. F : A -onto-> ~P A

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   {crab 2479   _Vcvv 2763   ~Pcpw 3605   ran crn 4664   Fn wfn 5253   -onto->wfo 5256   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by: (None)