Theorem avril1 30533

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 2014	. . . . . . . 8 ⊢ 𝑥 = 𝑥
2		dfnul2 4276	. . . . . . . . . 10 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
3	2	eqabri 2878	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
4	3	con2bii 357	. . . . . . . 8 ⊢ (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅)
5	1, 4	mpbi 230	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
6		eleq1 2824	. . . . . . 7 ⊢ (𝑥 = ⟨𝐹, 0⟩ → (𝑥 ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅))
7	5, 6	mtbii 326	. . . . . 6 ⊢ (𝑥 = ⟨𝐹, 0⟩ → ¬ ⟨𝐹, 0⟩ ∈ ∅)
8	7	vtocleg 3498	. . . . 5 ⊢ (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9		elex 3450	. . . . . 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 5086	. . . . 5 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13		0cn 11136	. . . . . . . 8 ⊢ 0 ∈ ℂ
14	13	mulridi 11149	. . . . . . 7 ⊢ (0 · 1) = 0
15	14	opeq2i 4820	. . . . . 6 ⊢ ⟨𝐹, (0 · 1)⟩ = ⟨𝐹, 0⟩
16	15	eleq1i 2827	. . . . 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 11047	. . . . . . . 8 ⊢ i = ⟨0R, 1R⟩
21	20	fveq1i 6841	. . . . . . 7 ⊢ (i‘1) = (⟨0R, 1R⟩‘1)
22		df-fv 6506	. . . . . . 7 ⊢ (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
23	21, 22	eqtri 2759	. . . . . 6 ⊢ (i‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
24	23	breq2i 5093	. . . . 5 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦))
25		df-r 11048	. . . . . . 7 ⊢ ℝ = (R × {0R})
26		sseq2 3948	. . . . . . . . 9 ⊢ (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R})))
27	26	abbidv 2802	. . . . . . . 8 ⊢ (ℝ = (R × {0R}) → {𝑧 ∣ 𝑧 ⊆ ℝ} = {𝑧 ∣ 𝑧 ⊆ (R × {0R})})
28		df-pw 4543	. . . . . . . 8 ⊢ 𝒫 ℝ = {𝑧 ∣ 𝑧 ⊆ ℝ}
29		df-pw 4543	. . . . . . . 8 ⊢ 𝒫 (R × {0R}) = {𝑧 ∣ 𝑧 ⊆ (R × {0R})}
30	27, 28, 29	3eqtr4g 2796	. . . . . . 7 ⊢ (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R}))
31	25, 30	ax-mp 5	. . . . . 6 ⊢ 𝒫 ℝ = 𝒫 (R × {0R})
32	31	breqi 5091	. . . . 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 625	. . 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 1542   ∈ wcel 2114  {cab 2714  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ⟨cop 4573   class class class wbr 5085   × cxp 5629  ℩cio 6452  ‘cfv 6498  (class class class)co 7367  Rcnr 10788  0_Rc0r 10789  1_Rc1r 10790  ℝcr 11037  0cc0 11038  1c1 11039  ici 11040   · cmul 11043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-mulcl 11100  ax-mulcom 11102  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1rid 11108  ax-cnre 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-i 11047  df-r 11048

This theorem is referenced by: (None)