Theorem canth 5112

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 5548. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 5113 for a counterexample. (Use nex 1456 if you want the form -. E.ff:A-onto->~PA.)

Hypothesis

Ref	Expression
canth.1	|- A e. _V

Assertion

Ref	Expression
canth	|- -. F:A-onto->~PA

Proof of Theorem canth

Step	Hyp	Ref	Expression
1		forn 4620	. 2 |- (F:A-onto->~PA -> ran F = ~PA)
2		fof 4617	. . 3 |- (F:A-onto->~PA -> F:A-->~PA)
3		id 73	. . . . . . . . . 10 |- (x = y -> x = y)
4		fveq2 4681	. . . . . . . . . 10 |- (x = y -> (F` x) = (F` y))
5	3, 4	eleq12d 1965	. . . . . . . . 9 |- (x = y -> (x e. (F` x) <-> y e. (F` y)))
6	5	notbid 673	. . . . . . . 8 |- (x = y -> (-. x e. (F` x) <-> -. y e. (F` y)))
7	6	elrab 2414	. . . . . . 7 |- (y e. {x e. A | -. x e. (F` x)} <-> (y e. A /\ -. y e. (F` y)))
8	7	baibr 750	. . . . . 6 |- (y e. A -> (-. y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}))
9		nbbn 724	. . . . . . 7 |- ((-. 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)}))
10		eleq2 1958	. . . . . . . 8 |- ((F` y) = {x e. A | -. x e. (F` x)} -> (y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}))
11	10	con3i 114	. . . . . . 7 |- (-. (y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}) -> -. (F` y) = {x e. A | -. x e. (F` x)})
12	9, 11	sylbi 216	. . . . . 6 |- ((-. y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}) -> -. (F` y) = {x e. A | -. x e. (F` x)})
13	8, 12	syl 12	. . . . 5 |- (y e. A -> -. (F` y) = {x e. A | -. x e. (F` x)})
14	13	rgen 2159	. . . 4 |- A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)}
15		ffn 4562	. . . . . . 7 |- (F:A-->~PA -> F Fn A)
16		fvelrnb 4719	. . . . . . . 8 |- (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)}))
17	16	biimpd 170	. . . . . . 7 |- (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)}))
18	15, 17	syl 12	. . . . . 6 |- (F:A-->~PA -> ({x e. A | -. x e. (F` x)} e. ran F -> E.y e. A (F` y) = {x e. A | -. x e. (F` x)}))
19	18	con3d 111	. . . . 5 |- (F:A-->~PA -> (-. E.y e. A (F` y) = {x e. A | -. x e. (F` x)} -> -. {x e. A | -. x e. (F` x)} e. ran F))
20		ralnex 2113	. . . . 5 |- (A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)} <-> -. E.y e. A (F` y) = {x e. A | -. x e. (F` x)})
21	19, 20	syl5ib 223	. . . 4 |- (F:A-->~PA -> (A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)} -> -. {x e. A | -. x e. (F` x)} e. ran F))
22	14, 21	mpi 55	. . 3 |- (F:A-->~PA -> -. {x e. A | -. x e. (F` x)} e. ran F)
23		ssrab2 2692	. . . . . 6 |- {x e. A | -. x e. (F` x)} C_ A
24		canth.1	. . . . . . . 8 |- A e. _V
25	24	rabex 3461	. . . . . . 7 |- {x e. A | -. x e. (F` x)} e. _V
26	25	elpw 3037	. . . . . 6 |- ({x e. A | -. x e. (F` x)} e. ~PA <-> {x e. A | -. x e. (F` x)} C_ A)
27	23, 26	mpbir 207	. . . . 5 |- {x e. A | -. x e. (F` x)} e. ~PA
28		eleq2 1958	. . . . 5 |- (ran F = ~PA -> ({x e. A | -. x e. (F` x)} e. ran F <-> {x e. A | -. x e. (F` x)} e. ~PA))
29	27, 28	mpbiri 211	. . . 4 |- (ran F = ~PA -> {x e. A | -. x e. (F` x)} e. ran F)
30	29	con3i 114	. . 3 |- (-. {x e. A | -. x e. (F` x)} e. ran F -> -. ran F = ~PA)
31	2, 22, 30	3syl 24	. 2 |- (F:A-onto->~PA -> -. ran F = ~PA)
32	1, 31	pm2.65i 150	1 |- -. F:A-onto->~PA

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  ran crn 3987   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998

This theorem is referenced by:  canth2 5548

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014

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