Theorem canthwdom 8367

Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7998, equivalent to canth 6508). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
canthwdom	⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Proof of Theorem canthwdom

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

Step	Hyp	Ref	Expression
1		0elpw 4760	. . . . 5 ⊢ ∅ ∈ 𝒫 𝐴
2		ne0i 3880	. . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
3	1, 2	mp1i 13	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅)
4		brwdomn0 8357	. . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
5	3, 4	syl 17	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴))
6	5	ibi 255	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
7		relwdom 8354	. . . . 5 ⊢ Rel ≼*
8	7	brrelex2i 5083	. . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V)
9		foeq2 6025	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥))
10		pweq 4111	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11		foeq3 6026	. . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
13	9, 12	bitrd 267	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴))
14	13	notbid 307	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴))
15		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
16	15	canth 6508	. . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥
17	14, 16	vtoclg 3239	. . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
18	8, 17	syl 17	. . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴)
19	18	nexdv 1851	. 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)
20	6, 19	pm2.65i 184	1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  –onto→wfo 5802   ≼^* cwdom 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-wdom 8347

This theorem is referenced by:  pwcdadom  8921