Theorem canth 3913

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 4490. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 3914 for a counterexample. (Use nex 1103 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 3680	. 2 |- (F:A-onto->P~A -> ran F = P~A)
2		fof 3678	. . 3 |- (F:A-onto->P~A -> F:A-->P~A)
3		id 59	. . . . . . . . . 10 |- (x = y -> x = y)
4		fveq2 3730	. . . . . . . . . 10 |- (x = y -> (F` x) = (F` y))
5	3, 4	eleq12d 1545	. . . . . . . . 9 |- (x = y -> (x e. (F` x) <-> y e. (F` y)))
6	5	negbid 613	. . . . . . . 8 |- (x = y -> (-. x e. (F` x) <-> -. y e. (F` y)))
7	6	elrab 1908	. . . . . . 7 |- (y e. {x e. A | -. x e. (F` x)} <-> (y e. A /\ -. y e. (F` y)))
8	7	baibr 688	. . . . . 6 |- (y e. A -> (-. y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}))
9		nbbn 663	. . . . . . 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 1538	. . . . . . . 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 199	. . . . . 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 1701	. . . 4 |- A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)}
15		ffn 3633	. . . . . . 7 |- (F:A-->P~A -> F Fn A)
16		fvelrnb 3766	. . . . . . . 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 153	. . . . . . 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 1656	. . . . 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 206	. . . 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 2134	. . . . . 6 |- {x e. A | -. x e. (F` x)} (_ A
24		canth.1	. . . . . . . 8 |- A e. V
25	24	rabex 2730	. . . . . . 7 |- {x e. A | -. x e. (F` x)} e. V
26	25	elpw 2408	. . . . . 6 |- ({x e. A | -. x e. (F` x)} e. P~A <-> {x e. A | -. x e. (F` x)} (_ A)
27	23, 26	mpbir 190	. . . . 5 |- {x e. A | -. x e. (F` x)} e. P~A
28		eleq2 1538	. . . . 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 194	. . . 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 135	1 |- -. F:A-onto->P~A

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649  {crab 1651  Vcvv 1814   (_ wss 2050  P~cpw 2405  ran crn 3177   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  ` cfv 3188

This theorem is referenced by:  canth2 4490

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204

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