Theorem canth2 9094

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 7341. 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 5335	. . 3 ⊢ 𝒫 𝐴 ∈ V
3		snelpwi 5403	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		vex 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	sneqr 4804	. . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6		sneq 4599	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	5, 6	impbii 209	. . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
8	7	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
9	3, 8	dom3 8967	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
10	1, 2, 9	mp2an 692	. 2 ⊢ 𝐴 ≼ 𝒫 𝐴
11	1	canth 7341	. . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴
12		f1ofo 6807	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴)
13	11, 12	mto 197	. . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴
14	13	nex 1800	. . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴
15		bren 8928	. . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴)
16	14, 15	mtbir 323	. 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴
17		brsdom 8946	. 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
18	10, 16, 17	mpbir2an 711	1 ⊢ 𝐴 ≺ 𝒫 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  –onto→wfo 6509  –1-1-onto→wf1o 6510   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-en 8919  df-dom 8920  df-sdom 8921

This theorem is referenced by:  canth2g  9095  r1sdom  9727  alephsucpw2  10064  dfac13  10096  pwsdompw  10156  numthcor  10447  alephexp1  10532  pwcfsdom  10536  cfpwsdom  10537  gchac  10634  inawinalem  10642  tskcard  10734  gruina  10771  grothac  10783  rpnnen  16195  rexpen  16196  rucALT  16198  rectbntr0  24721