Theorem canth 6036

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 1553 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 3333	. . . 4 |- { x e. A | -. x e. ( F ` x ) } C_ A
3	1, 2	elpwi2 4294	. . 3 |- { x e. A | -. x e. ( F ` x ) } e. ~P A
4		forn 5618	. . 3 |- ( F : A -onto-> ~P A -> ran F = ~P A )
5	3, 4	eleqtrrid 2328	. 2 |- ( F : A -onto-> ~P A -> { x e. A | -. x e. ( F ` x ) } e. ran F )
6		pm5.19 718	. . . . . 6 |- -. ( y e. ( F ` y ) <-> -. y e. ( F ` y ) )
7		eleq2 2302	. . . . . . 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 5695	. . . . . . . . . 10 |- ( x = y -> ( F ` x ) = ( F ` y ) )
10	8, 9	eleq12d 2309	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` x ) <-> y e. ( F ` y ) ) )
11	10	notbid 677	. . . . . . . 8 |- ( x = y -> ( -. x e. ( F ` x ) <-> -. y e. ( F ` y ) ) )
12	11	elrab3 2983	. . . . . . 7 |- ( y e. A -> ( y e. { x e. A | -. x e. ( F ` x ) } <-> -. y e. ( F ` y ) ) )
13	7, 12	sylan9bbr 467	. . . . . 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 672	. . . . 5 |- -. ( y e. A /\ ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
15	14	imnani 702	. . . 4 |- ( y e. A -> -. ( F ` y ) = { x e. A | -. x e. ( F ` x ) } )
16	15	nrex 2642	. . 3 |- -. E. y e. A ( F ` y ) = { x e. A | -. x e. ( F ` x ) }
17		fofn 5617	. . . 4 |- ( F : A -onto-> ~P A -> F Fn A )
18		fvelrnb 5750	. . . 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 686	. 2 |- ( F : A -onto-> ~P A -> -. { x e. A | -. x e. ( F ` x ) } e. ran F )
21	5, 20	pm2.65i 648	1 |- -. F : A -onto-> ~P A

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   ~Pcpw 3688   ran crn 4775   Fn wfn 5372   -onto->wfo 5375   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385

This theorem is used by: (None)