Theorem canth2 9130

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 7362. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)

Hypothesis

Ref	Expression
canth2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
canth2	⊢ 𝐴 ≺ 𝒫 𝐴

Proof of Theorem canth2

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		canth2.1	. . 3 ⊢ 𝐴 ∈ V
2	1	pwex 5379	. . 3 ⊢ 𝒫 𝐴 ∈ V
3		snelpwi 5444	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	sneqr 4842	. . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6		sneq 4639	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	5, 6	impbii 208	. . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
8	7	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
9	3, 8	dom3 8992	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
10	1, 2, 9	mp2an 691	. 2 ⊢ 𝐴 ≼ 𝒫 𝐴
11	1	canth 7362	. . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴
12		f1ofo 6841	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴)
13	11, 12	mto 196	. . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴
14	13	nex 1803	. . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴
15		bren 8949	. . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴)
16	14, 15	mtbir 323	. 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴
17		brsdom 8971	. 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
18	10, 16, 17	mpbir2an 710	1 ⊢ 𝐴 ≺ 𝒫 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  –onto→wfo 6542  –1-1-onto→wf1o 6543   ≈ cen 8936   ≼ cdom 8937   ≺ csdm 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-en 8940  df-dom 8941  df-sdom 8942

This theorem is referenced by:  canth2g  9131  r1sdom  9769  alephsucpw2  10106  dfac13  10137  pwsdompw  10199  numthcor  10489  alephexp1  10574  pwcfsdom  10578  cfpwsdom  10579  gchac  10676  inawinalem  10684  tskcard  10776  gruina  10813  grothac  10825  rpnnen  16170  rexpen  16171  rucALT  16173  rectbntr0  24348