Theorem avril1 30392

Description: Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object 𝑥 equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at mmnotes.txt, under the 1-Apr-2006 entry.

Assertion

Ref	Expression
avril1	⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))

Proof of Theorem avril1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 2012	. . . . . . . 8 ⊢ 𝑥 = 𝑥
2		dfnul2 4299	. . . . . . . . . 10 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
3	2	eqabri 2871	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
4	3	con2bii 357	. . . . . . . 8 ⊢ (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅)
5	1, 4	mpbi 230	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
6		eleq1 2816	. . . . . . 7 ⊢ (𝑥 = ⟨𝐹, 0⟩ → (𝑥 ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅))
7	5, 6	mtbii 326	. . . . . 6 ⊢ (𝑥 = ⟨𝐹, 0⟩ → ¬ ⟨𝐹, 0⟩ ∈ ∅)
8	7	vtocleg 3519	. . . . 5 ⊢ (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9		elex 3468	. . . . . 6 ⊢ (⟨𝐹, 0⟩ ∈ ∅ → ⟨𝐹, 0⟩ ∈ V)
10	9	con3i 154	. . . . 5 ⊢ (¬ ⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
11	8, 10	pm2.61i 182	. . . 4 ⊢ ¬ ⟨𝐹, 0⟩ ∈ ∅
12		df-br 5108	. . . . 5 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13		0cn 11166	. . . . . . . 8 ⊢ 0 ∈ ℂ
14	13	mulridi 11178	. . . . . . 7 ⊢ (0 · 1) = 0
15	14	opeq2i 4841	. . . . . 6 ⊢ ⟨𝐹, (0 · 1)⟩ = ⟨𝐹, 0⟩
16	15	eleq1i 2819	. . . . 5 ⊢ (⟨𝐹, (0 · 1)⟩ ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅)
17	12, 16	bitri 275	. . . 4 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, 0⟩ ∈ ∅)
18	11, 17	mtbir 323	. . 3 ⊢ ¬ 𝐹∅(0 · 1)
19	18	intnan 486	. 2 ⊢ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1))
20		df-i 11077	. . . . . . . 8 ⊢ i = ⟨0R, 1R⟩
21	20	fveq1i 6859	. . . . . . 7 ⊢ (i‘1) = (⟨0R, 1R⟩‘1)
22		df-fv 6519	. . . . . . 7 ⊢ (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
23	21, 22	eqtri 2752	. . . . . 6 ⊢ (i‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
24	23	breq2i 5115	. . . . 5 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦))
25		df-r 11078	. . . . . . 7 ⊢ ℝ = (R × {0R})
26		sseq2 3973	. . . . . . . . 9 ⊢ (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R})))
27	26	abbidv 2795	. . . . . . . 8 ⊢ (ℝ = (R × {0R}) → {𝑧 ∣ 𝑧 ⊆ ℝ} = {𝑧 ∣ 𝑧 ⊆ (R × {0R})})
28		df-pw 4565	. . . . . . . 8 ⊢ 𝒫 ℝ = {𝑧 ∣ 𝑧 ⊆ ℝ}
29		df-pw 4565	. . . . . . . 8 ⊢ 𝒫 (R × {0R}) = {𝑧 ∣ 𝑧 ⊆ (R × {0R})}
30	27, 28, 29	3eqtr4g 2789	. . . . . . 7 ⊢ (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R}))
31	25, 30	ax-mp 5	. . . . . 6 ⊢ 𝒫 ℝ = 𝒫 (R × {0R})
32	31	breqi 5113	. . . . 5 ⊢ (𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦))
33	24, 32	bitri 275	. . . 4 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦))
34	33	anbi1i 624	. . 3 ⊢ ((𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1)))
35	34	notbii 320	. 2 ⊢ (¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1)))
36	19, 35	mpbir 231	1 ⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ⟨cop 4595   class class class wbr 5107   × cxp 5636  ℩cio 6462  ‘cfv 6511  (class class class)co 7387  Rcnr 10818  0_Rc0r 10819  1_Rc1r 10820  ℝcr 11067  0cc0 11068  1c1 11069  ici 11070   · cmul 11073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-mulcl 11130  ax-mulcom 11132  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1rid 11138  ax-cnre 11141

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-i 11077  df-r 11078

This theorem is referenced by: (None)