Theorem canth2 9119

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 7366. 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 5353	. . 3 ⊢ 𝒫 𝐴 ∈ V
3		snelpwi 5427	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	sneqr 4806	. . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6		sneq 4600	. . . . . 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 8994	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
10	1, 2, 9	mp2an 704	. 2 ⊢ 𝐴 ≼ 𝒫 𝐴
11	1	canth 7366	. . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴
12		f1ofo 6830	. . . . 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 1830	. . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴
15		bren 8954	. . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴)
16	14, 15	mtbir 326	. 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴
17		brsdom 8972	. 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
18	10, 16, 17	mpbir2an 723	1 ⊢ 𝐴 ≺ 𝒫 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  𝒫 cpw 4563  {csn 4590   class class class wbr 5110  –onto→wfo 6536  –1-1-onto→wf1o 6537   ≈ cen 8941   ≼ cdom 8942   ≺ csdm 8943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 8945  df-dom 8946  df-sdom 8947

This theorem is referenced by:  canth2g  9120  r1sdom  9747  alephsucpw2  10096  dfac13  10127  pwsdompw  10187  numthcor  10479  alephexp1  10565  pwcfsdom  10569  cfpwsdom  10570  gchac  10667  inawinalem  10675  tskcard  10767  gruina  10804  grothac  10816  rpnnen  16284  rexpen  16285  rucALT  16287  rectbntr0  24971