Theorem canth2 9152

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 7374. 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 5341	. . 3 ⊢ 𝒫 𝐴 ∈ V
3		snelpwi 5411	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	sneqr 4794	. . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6		sneq 4590	. . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
7	5, 6	impbii 212	. . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
8	7	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
9	3, 8	dom3 9026	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
10	1, 2, 9	mp2an 705	. 2 ⊢ 𝐴 ≼ 𝒫 𝐴
11	1	canth 7374	. . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴
12		f1ofo 6832	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴)
13	11, 12	mto 200	. . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴
14	13	nex 1832	. . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴
15		bren 8986	. . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴)
16	14, 15	mtbir 326	. 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴
17		brsdom 9004	. 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
18	10, 16, 17	mpbir2an 724	1 ⊢ 𝐴 ≺ 𝒫 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450  𝒫 cpw 4556  {csn 4584   class class class wbr 5102  –onto→wfo 6536  –1-1-onto→wf1o 6537   ≈ cen 8973   ≼ cdom 8974   ≺ csdm 8975

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-en 8977  df-dom 8978  df-sdom 8979

This theorem is used by:  canth2g  9153  r1sdom  9784  alephsucpw2  10193  dfac13  10224  pwsdompw  10284  numthcor  10575  alephexp1  10667  pwcfsdom  10671  cfpwsdom  10672  gchac  10769  inawinalem  10777  tskcard  10869  gruina  10906  grothac  10918  rpnnen  16400  rexpen  16401  rucALT  16403  rectbntr0  25157