Theorem avril1 30795

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 2042	. . . . . . . 8 ⊢ 𝑥 = 𝑥
2		dfnul2 4290	. . . . . . . . . 10 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
3	2	eqabri 2905	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
4	3	con2bii 360	. . . . . . . 8 ⊢ (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅)
5	1, 4	mpbi 233	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
6		eleq1 2851	. . . . . . 7 ⊢ (𝑥 = ⟨𝐹, 0⟩ → (𝑥 ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅))
7	5, 6	mtbii 329	. . . . . 6 ⊢ (𝑥 = ⟨𝐹, 0⟩ → ¬ ⟨𝐹, 0⟩ ∈ ∅)
8	7	vtocleg 3522	. . . . 5 ⊢ (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9		elex 3476	. . . . . 6 ⊢ (⟨𝐹, 0⟩ ∈ ∅ → ⟨𝐹, 0⟩ ∈ V)
10	9	con3i 155	. . . . 5 ⊢ (¬ ⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
11	8, 10	pm2.61i 184	. . . 4 ⊢ ¬ ⟨𝐹, 0⟩ ∈ ∅
12		df-br 5111	. . . . 5 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13		0cn 11199	. . . . . . . 8 ⊢ 0 ∈ ℂ
14	13	mulridi 11214	. . . . . . 7 ⊢ (0 · 1) = 0
15	14	opeq2i 4843	. . . . . 6 ⊢ ⟨𝐹, (0 · 1)⟩ = ⟨𝐹, 0⟩
16	15	eleq1i 2854	. . . . 5 ⊢ (⟨𝐹, (0 · 1)⟩ ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅)
17	12, 16	bitri 278	. . . 4 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, 0⟩ ∈ ∅)
18	11, 17	mtbir 326	. . 3 ⊢ ¬ 𝐹∅(0 · 1)
19	18	intnan 491	. 2 ⊢ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1))
20		df-i 11110	. . . . . . . 8 ⊢ i = ⟨0R, 1R⟩
21	20	fveq1i 6884	. . . . . . 7 ⊢ (i‘1) = (⟨0R, 1R⟩‘1)
22		df-fv 6546	. . . . . . 7 ⊢ (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
23	21, 22	eqtri 2786	. . . . . 6 ⊢ (i‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
24	23	breq2i 5118	. . . . 5 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦))
25		df-r 11111	. . . . . . 7 ⊢ ℝ = (R × {0R})
26		sseq2 3964	. . . . . . . . 9 ⊢ (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R})))
27	26	abbidv 2829	. . . . . . . 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 2823	. . . . . . 7 ⊢ (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R}))
31	25, 30	ax-mp 5	. . . . . 6 ⊢ 𝒫 ℝ = 𝒫 (R × {0R})
32	31	breqi 5116	. . . . 5 ⊢ (𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦))
33	24, 32	bitri 278	. . . 4 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦))
34	33	anbi1i 635	. . 3 ⊢ ((𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1)))
35	34	notbii 323	. 2 ⊢ (¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R⟩𝑦) ∧ 𝐹∅(0 · 1)))
36	19, 35	mpbir 234	1 ⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590  ⟨cop 4596   class class class wbr 5110   × cxp 5661  ℩cio 6492  ‘cfv 6538  (class class class)co 7412  Rcnr 10851  0_Rc0r 10852  1_Rc1r 10853  ℝcr 11100  0cc0 11101  1c1 11102  ici 11103   · cmul 11106

This theorem was proved from 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-12 2213  ax-ext 2735  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-mulcl 11163  ax-mulcom 11165  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1rid 11171  ax-cnre 11174

This theorem 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-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-i 11110  df-r 11111

This theorem is referenced by: (None)