Theorem canth 5828

Description: No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1500 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)

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		canth.1	. . . 4 ⊢ 𝐴 ∈ V
2		ssrab2 3240	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴
3	1, 2	elpwi2 4158	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴
4		forn 5441	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
5	3, 4	eleqtrrid 2267	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
6		pm5.19 706	. . . . . 6 ⊢ ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦))
7		eleq2 2241	. . . . . . 7 ⊢ ((𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} → (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
8		id 19	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
9		fveq2 5515	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
10	8, 9	eleq12d 2248	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑦)))
11	10	notbid 667	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹‘𝑥) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
12	11	elrab3 2894	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
13	7, 12	sylan9bbr 463	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) → (𝑦 ∈ (𝐹‘𝑦) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
14	6, 13	mto 662	. . . . 5 ⊢ ¬ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})
15	14	imnani 691	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})
16	15	nrex 2569	. . 3 ⊢ ¬ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}
17		fofn 5440	. . . 4 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → 𝐹 Fn 𝐴)
18		fvelrnb 5563	. . . 4 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
19	17, 18	syl 14	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
20	16, 19	mtbiri 675	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
21	5, 20	pm2.65i 639	1 ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  {crab 2459  Vcvv 2737  𝒫 cpw 3575  ran crn 4627   Fn wfn 5211  –onto→wfo 5214  ‘cfv 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224

This theorem is referenced by: (None)