Theorem canth2 9177

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 7397. 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 5346	. . 3 ⊢ 𝒫 𝐴 ∈ V
3		snelpwi 5416	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	sneqr 4798	. . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6		sneq 4594	. . . . . 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 9051	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
10	1, 2, 9	mp2an 705	. 2 ⊢ 𝐴 ≼ 𝒫 𝐴
11	1	canth 7397	. . . . 5 ⊢ ¬ 𝑓:𝐴⟶onto𝒫 𝐴
12		f1ofo 6854	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto𝒫 𝐴 → 𝑓:𝐴⟶onto𝒫 𝐴)
13	11, 12	mto 200	. . . 4 ⊢ ¬ 𝑓:𝐴⟶1-1onto𝒫 𝐴
14	13	nex 1833	. . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴⟶1-1onto𝒫 𝐴
15		bren 9011	. . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝒫 𝐴)
16	14, 15	mtbir 326	. 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴
17		brsdom 9029	. 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 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  𝒫 cpw 4560  {csn 4588   class class class wbr 5106  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557   ≈ cen 8998   ≼ cdom 8999   ≺ csdm 9000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-en 9002  df-dom 9003  df-sdom 9004

This theorem is used by:  canth2g  9178  r1sdom  9808  alephsucpw2  10217  dfac13  10248  pwsdompw  10308  numthcor  10599  alephexp1  10691  pwcfsdom  10695  cfpwsdom  10696  gchac  10793  inawinalem  10801  tskcard  10893  gruina  10930  grothac  10942  rpnnen  16424  rexpen  16425  rucALT  16427  rectbntr0  25182