Theorem canth2 4464

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 3892.

Hypothesis

Ref	Expression
canth2.1	⊢ A ∈ V

Assertion

Ref	Expression
canth2	⊢ A ≺ ℘A

Proof of Theorem canth2

Step	Hyp	Ref	Expression
1		brsdom 4363	. 2 ⊢ (A ≺ ℘A ↔ (A ≼ ℘A ⋀ ¬ A ≈ ℘A))
2		canth2.1	. . 3 ⊢ A ∈ V
3		visset 1804	. . . . . 6 ⊢ x ∈ V
4	3	snelpw 2742	. . . . 5 ⊢ (x ∈ A ↔ {x} ∈ ℘A)
5	4	biimp 151	. . . 4 ⊢ (x ∈ A → {x} ∈ ℘A)
6	3	sneqr 2468	. . . . . 6 ⊢ ({x} = {y} → x = y)
7		sneq 2407	. . . . . 6 ⊢ (x = y → {x} = {y})
8	6, 7	impbi 157	. . . . 5 ⊢ ({x} = {y} ↔ x = y)
9	8	a1i 8	. . . 4 ⊢ ((x ∈ A ⋀ y ∈ A) → ({x} = {y} ↔ x = y))
10	5, 9	dom2 4386	. . 3 ⊢ (A ∈ V → A ≼ ℘A)
11	2, 10	ax-mp 7	. 2 ⊢ A ≼ ℘A
12	2	canth 3892	. . . . 5 ⊢ ¬ f:A–onto→℘A
13		f1ofo 3680	. . . . 5 ⊢ (f:A–1-1-onto→℘A → f:A–onto→℘A)
14	12, 13	mto 106	. . . 4 ⊢ ¬ f:A–1-1-onto→℘A
15	14	nex 1097	. . 3 ⊢ ¬ ∃f f:A–1-1-onto→℘A
16	2	pwex 2735	. . . 4 ⊢ ℘A ∈ V
17	16	bren 4359	. . 3 ⊢ (A ≈ ℘A ↔ ∃f f:A–1-1-onto→℘A)
18	15, 17	mtbir 192	. 2 ⊢ ¬ A ≈ ℘A
19	1, 11, 18	mpbir2an 728	1 ⊢ A ≺ ℘A

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∃wex 977  Vcvv 1802  ℘cpw 2391  {csn 2399   class class class wbr 2609  –onto→wfo 3170  –1-1-onto→wf1o 3171   ≈ cen 4348   ≼ cdom 4349   ≺ csdm 4350

This theorem is referenced by:  2pwuninel 4465  canth2g 4466  1sdom2 4505  numthcor 4758  alephsucpw 4842  pnfnemnf 5509  infmap1 7516

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-9 962  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-rep 2683  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-rel 3175  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-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-en 4351  df-dom 4352  df-sdom 4353

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