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Theorem canth 3898

Description: No set is equinumerous to its power set (Cantor's theorem), i.e. no function can map it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 4470. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 3899 for a counterexample. (Use nex 1099 if you want the form .)

Hypothesis

Ref	Expression
canth.1

Assertion

Ref	Expression
canth

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Proof of Theorem canth

Step	Hyp	Ref	Expression
1		forn 3665	. 2
2		fof 3663	. . 3
3		id 59	. . . . . . . . . 10
4		fveq2 3715	. . . . . . . . . 10
5	3, 4	eleq12d 1539	. . . . . . . . 9
6	5	negbid 610	. . . . . . . 8
7	6	elrab 1901	. . . . . . 7
8	7	baibr 685	. . . . . 6
9		nbbn 660	. . . . . . 7
10		eleq2 1532	. . . . . . . 8
11	10	con3i 98	. . . . . . 7
12	9, 11	sylbi 199	. . . . . 6
13	8, 12	syl 10	. . . . 5
14	13	rgen 1695	. . . 4
15		ffn 3619	. . . . . . 7
16		fvelrnb 3751	. . . . . . . 8
17	16	biimpd 153	. . . . . . 7
18	15, 17	syl 10	. . . . . 6
19	18	con3d 95	. . . . 5
20		ralnex 1650	. . . . 5
21	19, 20	syl5ib 206	. . . 4
22	14, 21	mpi 44	. . 3
23		ssrab2 2127	. . . . . 6
24		canth.1	. . . . . . . 8
25	24	rabex 2720	. . . . . . 7
26	25	elpw 2400	. . . . . 6
27	23, 26	mpbir 190	. . . . 5
28		eleq2 1532	. . . . 5
29	27, 28	mpbiri 194	. . . 4
30	29	con3i 98	. . 3
31	2, 22, 30	3syl 20	. 2
32	1, 31	pm2.65i 135	1

Colors of variables: wff set class

Syntax hints:   wn 2   wi 3   wb 146   wceq 954   wcel 956  wral 1642  wrex 1643  crab 1645  cvv 1807   wss 2043  cpw 2397   crn 3166   wfn 3172  wf 3173  wfo 3175  cfv 3177

This theorem is referenced by:  canth2 4470

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193

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