Theorem canth2 4484

Description: Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 3907.

Hypothesis

Ref	Expression
canth2.1	|- A e. V

Assertion

Ref	Expression
canth2	|- A ~< P~A

Proof of Theorem canth2

Step	Hyp	Ref	Expression
1		brsdom 4381	. 2 |- (A ~< P~A <-> (A ~<_ P~A /\ -. A ~~ P~A))
2		canth2.1	. . 3 |- A e. V
3		visset 1813	. . . . . 6 |- x e. V
4	3	snelpw 2752	. . . . 5 |- (x e. A <-> {x} e. P~A)
5	4	biimp 151	. . . 4 |- (x e. A -> {x} e. P~A)
6	3	sneqr 2477	. . . . . 6 |- ({x} = {y} -> x = y)
7		sneq 2417	. . . . . 6 |- (x = y -> {x} = {y})
8	6, 7	impbi 157	. . . . 5 |- ({x} = {y} <-> x = y)
9	8	a1i 8	. . . 4 |- ((x e. A /\ y e. A) -> ({x} = {y} <-> x = y))
10	5, 9	dom2 4405	. . 3 |- (A e. V -> A ~<_ P~A)
11	2, 10	ax-mp 7	. 2 |- A ~<_ P~A
12	2	canth 3907	. . . . 5 |- -. f:A-onto->P~A
13		f1ofo 3695	. . . . 5 |- (f:A-1-1-onto->P~A -> f:A-onto->P~A)
14	12, 13	mto 106	. . . 4 |- -. f:A-1-1-onto->P~A
15	14	nex 1101	. . 3 |- -. E.f f:A-1-1-onto->P~A
16	2	pwex 2745	. . . 4 |- P~A e. V
17	16	bren 4377	. . 3 |- (A ~~ P~A <-> E.f f:A-1-1-onto->P~A)
18	15, 17	mtbir 192	. 2 |- -. A ~~ P~A
19	1, 11, 18	mpbir2an 730	1 |- A ~< P~A

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811  P~cpw 2401  {csn 2409   class class class wbr 2619  -onto->wfo 3180  -1-1-onto->wf1o 3181   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  canth2g 4485  1sdom2 4526  numthcor 4786  alephsucpw 4870  pnfnemnf 5536  infmap1 7573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-dom 4369  df-sdom 4370

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