Theorem canth 7364

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. For the equinumerosity version, see canth2 9114. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7365 for a counterexample. (Use nex 1830 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 4034	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴
3	1, 2	elpwi2 5306	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴
4		forn 6795	. . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
5	3, 4	eleqtrrid 2870	. 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹)
6		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
8	6, 7	eleq12d 2857	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑦)))
9	8	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹‘𝑥) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)))
10	9	elrab 3650	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (𝐹‘𝑦)))
11	10	baibr 545	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
12		nbbn 386	. . . . . 6 ⊢ ((¬ 𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) ↔ ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
13	11, 12	sylib 221	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
14		eleq2 2852	. . . . 5 ⊢ ((𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} → (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
15	13, 14	nsyl 141	. . . 4 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})
16	15	nrex 3093	. . 3 ⊢ ¬ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}
17		fofn 6794	. . . 4 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → 𝐹 Fn 𝐴)
18		fvelrnb 6941	. . . 4 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}))
19	17, 18	syl 18	. . 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 330	. 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

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455  𝒫 cpw 4562  ran crn 5662   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536

This proof depends on 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 5257  ax-nul 5269  ax-pr 5404

This proof 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-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544

This theorem is used by:  canth2  9114  canthwdom  9537