Theorem canth 3892

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 4464. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 3893 for a counterexample. (Use nex 1097 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 3659	. 2 |- (F:A-onto->P~A -> ran F = P~A)
2		fof 3657	. . 3 |- (F:A-onto->P~A -> F:A-->P~A)
3		id 59	. . . . . . . . . 10 |- (x = y -> x = y)
4		fveq2 3709	. . . . . . . . . 10 |- (x = y -> (F` x) = (F` y))
5	3, 4	eleq12d 1534	. . . . . . . . 9 |- (x = y -> (x e. (F` x) <-> y e. (F` y)))
6	5	negbid 609	. . . . . . . 8 |- (x = y -> (-. x e. (F` x) <-> -. y e. (F` y)))
7	6	elrab 1896	. . . . . . 7 |- (y e. {x e. A | -. x e. (F` x)} <-> (y e. A /\ -. y e. (F` y)))
8	7	baibr 684	. . . . . 6 |- (y e. A -> (-. y e. (F` y) <-> y e. {x e. A | -. x e. (F` x)}))
9		nbbn 659	. . . . . . 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 1527	. . . . . . . 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 1690	. . . 4 |- A.y e. A -. (F` y) = {x e. A | -. x e. (F` x)}
15		ffn 3613	. . . . . . 7 |- (F:A-->P~A -> F Fn A)
16		fvelrnb 3745	. . . . . . . 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 1645	. . . . 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 2121	. . . . . 6 |- {x e. A | -. x e. (F` x)} (_ A
24		canth.1	. . . . . . . 8 |- A e. V
25	24	rabex 2715	. . . . . . 7 |- {x e. A | -. x e. (F` x)} e. V
26	25	elpw 2394	. . . . . 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 1527	. . . . 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 953   e. wcel 955  A.wral 1637  E.wrex 1638  {crab 1640  Vcvv 1802   (_ wss 2037  P~cpw 2391  ran crn 3161   Fn wfn 3167  -->wf 3168  -onto->wfo 3170  ` cfv 3172

This theorem is referenced by:  canth2 4464

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188

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