Theorem canth 6648

Description: No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e. no function can map 𝐴 it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8154. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6649 for a counterexample. (Use nex 1771 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Hypothesis

Ref	Expression
canth.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
canth	⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴

Proof of Theorem canth

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

Step	Hyp	Ref	Expression
1		ssrab2 3720	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴
2		canth.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elpw2 4858	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴)
4	1, 3	mpbir 221	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴
5		forn 6156	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
6	4, 5	syl5eleqr 2737	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
7		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
8		fveq2 6229	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
9	7, 8	eleq12d 2724	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑦)))
10	9	notbid 307	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹‘𝑥) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
11	10	elrab 3396	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐹‘𝑦)))
12	11	baibr 965	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
13		nbbn 372	. . . . . 6 ⊢ ((¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) ↔ ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
14	12, 13	sylib 208	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
15		eleq2 2719	. . . . 5 ⊢ ((𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} → (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
16	14, 15	nsyl 135	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})
17	16	nrex 3029	. . 3 ⊢ ¬ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}
18		fofn 6155	. . . 4 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → 𝐹 Fn 𝐴)
19		fvelrnb 6282	. . . 4 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
20	18, 19	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
21	17, 20	mtbiri 316	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
22	6, 21	pm2.65i 185	1 ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  ran crn 5144   Fn wfn 5921  –onto→wfo 5924  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934

This theorem is referenced by:  canth2  8154  canthwdom  8525