Theorem canth2 3381

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 2945.

Hypothesis

Ref	Expression
canth2.1	|- A e. V

Assertion

Ref	Expression
canth2	|- A ~< P~A

Proof of Theorem canth2

Step	Hyp	Ref	Expression
1		canth2.1	. . . 4 |- A e. V
2		visset 1350	. . . . . . 7 |- x e. V
3	2	snelpw 1861	. . . . . 6 |- (x e. A <-> {x} e. P~A)
4	3	biimp 133	. . . . 5 |- (x e. A -> {x} e. P~A)
5	2	sneqr 1856	. . . . . . 7 |- ({x} = {y} -> x = y)
6		sneq 1816	. . . . . . 7 |- (x = y -> {x} = {y})
7	5, 6	impbi 139	. . . . . 6 |- ({x} = {y} <-> x = y)
8	7	a1i 7	. . . . 5 |- ((x e. A /\ y e. A) -> ({x} = {y} <-> x = y))
9	4, 8	dom2 3308	. . . 4 |- (A e. V -> A ~<_ P~A)
10	1, 9	ax-mp 6	. . 3 |- A ~<_ P~A
11	1	canth 2945	. . . . . 6 |- -. f:A-onto->P~A
12		df-f1o 2437	. . . . . . 7 |- (f:A-1-1-onto->P~A <-> (f:A-1-1->P~A /\ f:A-onto->P~A))
13	12	pm3.27bd 263	. . . . . 6 |- (f:A-1-1-onto->P~A -> f:A-onto->P~A)
14	11, 13	mto 93	. . . . 5 |- -. f:A-1-1-onto->P~A
15	14	nex 779	. . . 4 |- -. E.f f:A-1-1-onto->P~A
16	1	pwex 1806	. . . . 5 |- P~A e. V
17	16	bren 3282	. . . 4 |- (A ~~ P~A <-> E.f f:A-1-1-onto->P~A)
18	15, 17	mtbir 167	. . 3 |- -. A ~~ P~A
19	10, 18	pm3.2i 234	. 2 |- (A ~<_ P~A /\ -. A ~~ P~A)
20		brsdom 3286	. 2 |- (A ~< P~A <-> (A ~<_ P~A /\ -. A ~~ P~A))
21	19, 20	mpbir 165	1 |- A ~< P~A

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  Vcvv 1348  P~cpw 1798  {csn 1808   class class class wbr 2054  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  canth2g 3382  numthcor 3601  aleph1 3676  infmap1 4950

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-dom 3275  df-sdom 3276

metamath.org