Theorem canth 5790

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 1487 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 3222	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
3	1, 2	elpwi2 4131	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
4		forn 5407	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
5	3, 4	eleqtrrid 2254	. 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 2228	. . . . . . 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 5480	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
10	8, 9	eleq12d 2235	. . . . . . . . 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 2878	. . . . . . 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 2556	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
17		fofn 5406	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
18		fvelrnb 5528	. . . 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 1342   e. wcel 2135   E.wrex 2443   {crab 2446   _Vcvv 2721   ~Pcpw 3553   ran crn 4599   Fn wfn 5177   -onto->wfo 5180   ` cfv 5182

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fo 5188  df-fv 5190

This theorem is referenced by: (None)