Theorem canth 4205

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 4629. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 4206 for a counterexample. (Use nex 1137 if you want the form -. E.ff:A-onto->P~A.)

Hypothesis

Ref	Expression
canth.1	|- A e. V

Assertion

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

Proof of Theorem canth

Step	Hyp	Ref	Expression
1		forn 3782	. 2 |- (F:A-onto->P~A -> ran F = P~A)
2		fof 3779	. . 3 |- (F:A-onto->P~A -> F:A-->P~A)
3		id 59	. . . . . . . . . 10 |- (x = y -> x = y)
4		fveq2 3835	. . . . . . . . . 10 |- (x = y -> (F` x) = (F` y))
5	3, 4	eleq12d 1585	. . . . . . . . 9 |- (x = y -> (x e. (F` x) <-> y e. (F` y)))
6	5	notbid 614	. . . . . . . 8 |- (x = y -> (-. x e. (F` x) <-> -. y e. (F` y)))
7	6	elrab 1951	. . . . . . 7 |- (y e. {x e. A | -. x e. (F` x)} <-> (y e. A /\ -. y e. (F` y)))
8	7	baibr 690	. . . . . 6 |- (y e. A -> (-. y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}))
9		nbbn 664	. . . . . . 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 1578	. . . . . . . 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 98	. . . . . . 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 197	. . . . . 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 10	. . . . 5 |- (y e. A -> -. (F` y) = {x e. A | -. x e. (F` x)})
14	13	rgen 1744	. . . 4 |- A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)}
15		ffn 3734	. . . . . . 7 |- (F:A-->P~A -> F Fn A)
16		fvelrnb 3871	. . . . . . . 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 151	. . . . . . 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 10	. . . . . 6 |- (F:A-->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)}))
19	18	con3d 95	. . . . 5 |- (F:A-->P~A -> (-. 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 1699	. . . . 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 204	. . . 4 |- (F:A-->P~A -> (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 44	. . 3 |- (F:A-->P~A -> -. {x e. A | -. x e. (F` x)} e. ran F)
23		ssrab2 2183	. . . . . 6 |- {x e. A | -. x e. (F` x)} (_ A
24		canth.1	. . . . . . . 8 |- A e. V
25	24	rabex 2799	. . . . . . 7 |- {x e. A | -. x e. (F` x)} e. V
26	25	elpw 2461	. . . . . 6 |- ({x e. A | -. x e. (F` x)} e. P~A <-> {x e. A | -. x e. (F` x)} (_ A)
27	23, 26	mpbir 188	. . . . 5 |- {x e. A | -. x e. (F` x)} e. P~A
28		eleq2 1578	. . . . 5 |- (ran F = P~A -> ({x e. A | -. x e. (F` x)} e. ran F <-> {x e. A | -. x e. (F` x)} e. P~A))
29	27, 28	mpbiri 192	. . . 4 |- (ran F = P~A -> {x e. A | -. x e. (F` x)} e. ran F)
30	29	con3i 98	. . 3 |- (-. {x e. A | -. x e. (F` x)} e. ran F -> -. ran F = P~A)
31	2, 22, 30	3syl 20	. 2 |- (F:A-onto->P~A -> -. ran F = P~A)
32	1, 31	pm2.65i 133	1 |- -. F:A-onto->P~A

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692  {crab 1694  Vcvv 1857   (_ wss 2099  P~cpw 2458  ran crn 3252   Fn wfn 3258  -->wf 3259  -onto->wfo 3261  ` cfv 3263

This theorem is referenced by:  canth2 4629

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-fo 3277  df-fv 3279

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