Theorem canth 5807

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 1493 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 3232	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
3	1, 2	elpwi2 4144	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
4		forn 5423	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
5	3, 4	eleqtrrid 2260	. 2 |- ( F : A -onto-> ~P A -> { x e. A | -. x e. ( F ` x ) } e. ran F )
6		pm5.19 701	. . . . . 6 |- -. ( y e. ( F ` y ) <-> -. y e. ( F ` y ) )
7		eleq2 2234	. . . . . . 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 5496	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
10	8, 9	eleq12d 2241	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` x ) <-> y e. ( F ` y ) ) )
11	10	notbid 662	. . . . . . . 8 |- ( x = y -> ( -. x e. ( F ` x ) <-> -. y e. ( F ` y ) ) )
12	11	elrab3 2887	. . . . . . 7 |- ( y e. A -> ( y e. { x e. A | -. x e. ( F ` x ) } <-> -. y e. ( F ` y ) ) )
13	7, 12	sylan9bbr 460	. . . . . 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 657	. . . . 5 |- -. ( y e. A /\ ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
15	14	imnani 686	. . . 4 |- ( y e. A -> -. ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
16	15	nrex 2562	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
17		fofn 5422	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
18		fvelrnb 5544	. . . 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 670	. 2 |- ( F : A -onto-> ~P A -> -. { x e. A | -. x e. ( F ` x ) } e. ran F )
21	5, 20	pm2.65i 634	1 |- -. F : A -onto-> ~P A

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   ~Pcpw 3566   ran crn 4612   Fn wfn 5193   -onto->wfo 5196   ` cfv 5198

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206

This theorem is referenced by: (None)