Theorem canth 2945

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 3381. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 2946 for a counterexample.

Hypothesis

Ref	Expression
canth.1	⊢ A ∈ V

Assertion

Ref	Expression
canth	⊢ ¬ F:A–onto→℘A

Proof of Theorem canth

Step	Hyp	Ref	Expression
1		forn 2789	. 2 ⊢ (F:A–onto→℘A → ran F = ℘A)
2		fof 2788	. . 3 ⊢ (F:A–onto→℘A → F:A–→℘A)
3		id 9	. . . . . . . . . 10 ⊢ (x = y → x = y)
4		fveq2 2832	. . . . . . . . . 10 ⊢ (x = y → (F ‘x) = (F ‘y))
5	3, 4	eleq12d 1157	. . . . . . . . 9 ⊢ (x = y → (x ∈ (F ‘x) ↔ y ∈ (F ‘y)))
6	5	negbid 463	. . . . . . . 8 ⊢ (x = y → (¬ x ∈ (F ‘x) ↔ ¬ y ∈ (F ‘y)))
7	6	elrab 1422	. . . . . . 7 ⊢ (y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)} ↔ (y ∈ A ∧ ¬ y ∈ (F ‘y)))
8	7	baibr 507	. . . . . 6 ⊢ (y ∈ A → (¬ y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}))
9		nbbn 498	. . . . . . 7 ⊢ ((¬ y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}) ↔ ¬ (y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}))
10		eleq2 1150	. . . . . . . 8 ⊢ ((F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)} → (y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}))
11	10	con3i 90	. . . . . . 7 ⊢ (¬ (y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}) → ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)})
12	9, 11	sylbi 174	. . . . . 6 ⊢ ((¬ y ∈ (F ‘y) ↔ y ∈ {x ∈ A∣ ¬ x ∈ (F ‘x)}) → ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)})
13	8, 12	syl 12	. . . . 5 ⊢ (y ∈ A → ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)})
14	13	rgen 1247	. . . 4 ⊢ ∀y ∈ A ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)}
15		ffn 2752	. . . . . . 7 ⊢ (F:A–→℘A → F Fn A)
16		fvelrn 2883	. . . . . . . 8 ⊢ (F Fn A → ({x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F ↔ ∃y ∈ A (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)}))
17	16	biimpd 135	. . . . . . 7 ⊢ (F Fn A → ({x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F → ∃y ∈ A (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)}))
18	15, 17	syl 12	. . . . . 6 ⊢ (F:A–→℘A → ({x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F → ∃y ∈ A (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)}))
19	18	con3d 87	. . . . 5 ⊢ (F:A–→℘A → (¬ ∃y ∈ A (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)} → ¬ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F))
20		ralnex 1209	. . . . 5 ⊢ (∀y ∈ A ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)} ↔ ¬ ∃y ∈ A (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)})
21	19, 20	syl5ib 181	. . . 4 ⊢ (F:A–→℘A → (∀y ∈ A ¬ (F ‘y) = {x ∈ A∣ ¬ x ∈ (F ‘x)} → ¬ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F))
22	14, 21	mpi 44	. . 3 ⊢ (F:A–→℘A → ¬ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F)
23		ssrab 1556	. . . . . 6 ⊢ {x ∈ A∣ ¬ x ∈ (F ‘x)} ⊆ A
24		canth.1	. . . . . . . 8 ⊢ A ∈ V
25	24	rabex 1706	. . . . . . 7 ⊢ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ V
26	25	elpw 1801	. . . . . 6 ⊢ ({x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ℘A ↔ {x ∈ A∣ ¬ x ∈ (F ‘x)} ⊆ A)
27	23, 26	mpbir 165	. . . . 5 ⊢ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ℘A
28		eleq2 1150	. . . . 5 ⊢ (ran F = ℘A → ({x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F ↔ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ℘A))
29	27, 28	mpbiri 169	. . . 4 ⊢ (ran F = ℘A → {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F)
30	29	con3i 90	. . 3 ⊢ (¬ {x ∈ A∣ ¬ x ∈ (F ‘x)} ∈ ran F → ¬ ran F = ℘A)
31	2, 22, 30	3syl 21	. 2 ⊢ (F:A–onto→℘A → ¬ ran F = ℘A)
32	1, 31	pm2.65i 116	1 ⊢ ¬ F:A–onto→℘A

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  ran crn 2411   Fn wfn 2417  –→wf 2418  –onto→wfo 2420   ‘cfv 2422

This theorem is referenced by:  canth2 3381

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fo 2436  df-fv 2438

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