Theorem canth 6608

Description: No set A is equinumerous to its power set (Cantor's theorem), i.e. no function can map A it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8113. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 6609 for a counterexample. (Use nex 1731 if you want the form -. E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

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		ssrab2 3687	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
2		canth.1	. . . . 5 |- A e. _V
3	2	elpw2 4828	. . . 4 |- ( { x e. A | -. x e. ( F ` x ) } e. ~P A <-> { x e. A | -. x e. ( F ` x ) } C_ A )
4	1, 3	mpbir 221	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
5		forn 6118	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
6	4, 5	syl5eleqr 2708	. 2 |- ( F : A -onto-> ~P A -> { x e. A | -. x e. ( F ` x ) } e. ran F )
7		id 22	. . . . . . . . . 10 |- ( x = y -> x = y )
8		fveq2 6191	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
9	7, 8	eleq12d 2695	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` x ) <-> y e. ( F ` y ) ) )
10	9	notbid 308	. . . . . . . 8 |- ( x = y -> ( -. x e. ( F ` x ) <-> -. y e. ( F ` y ) ) )
11	10	elrab 3363	. . . . . . 7 |- ( y e. { x e. A | -. x e. ( F ` x ) } <-> ( y e. A /\ -. y e. ( F ` y ) ) )
12	11	baibr 945	. . . . . 6 |- ( y e. A -> ( -. y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) )
13		nbbn 373	. . . . . 6 |- ( ( -. y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) <-> -. ( y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) )
14	12, 13	sylib 208	. . . . 5 |- ( y e. A -> -. ( y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) )
15		eleq2 2690	. . . . 5 |- ( ( F ` y ) = { x e. A | -. x e. ( F ` x ) } -> ( y e. ( F ` y ) <-> y e. { x e. A | -. x e. ( F ` x ) } ) )
16	14, 15	nsyl 135	. . . 4 |- ( y e. A -> -. ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
17	16	nrex 3000	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
18		fofn 6117	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
19		fvelrnb 6243	. . . 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 ) } ) )
20	18, 19	syl 17	. . 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 ) } ) )
21	17, 20	mtbiri 317	. 2 |- ( F : A -onto-> ~P A -> -. { x e. A | -. x e. ( F ` x ) } e. ran F )
22	6, 21	pm2.65i 185	1 |- -. F : A -onto-> ~P A

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  canth2  8113  canthwdom  8484