Theorem canth 7113

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 8672. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7114 for a counterexample. (Use nex 1801 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		canth.1	. . . 4 ⊢ 𝐴 ∈ V
2		ssrab2 4058	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴
3	1, 2	elpwi2 5251	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴
4		forn 6595	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
5	3, 4	eleqtrrid 2922	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
6		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eleq12d 2909	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑦)))
9	8	notbid 320	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹‘𝑥) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
10	9	elrab 3682	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐹‘𝑦)))
11	10	baibr 539	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
12		nbbn 387	. . . . . 6 ⊢ ((¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) ↔ ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
13	11, 12	sylib 220	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
14		eleq2 2903	. . . . 5 ⊢ ((𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} → (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
15	13, 14	nsyl 142	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})
16	15	nrex 3271	. . 3 ⊢ ¬ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}
17		fofn 6594	. . . 4 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → 𝐹 Fn 𝐴)
18		fvelrnb 6728	. . . 4 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
19	17, 18	syl 17	. . 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 329	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
21	5, 20	pm2.65i 196	1 ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  {crab 3144  Vcvv 3496  𝒫 cpw 4541  ran crn 5558   Fn wfn 6352  –onto→wfo 6355  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365

This theorem is referenced by:  canth2  8672  canthwdom  9045