Theorem avril1 30425

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 4289	. . . . . . . . . 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 3510	. . . . 5 ⊢ (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9		elex 3459	. . . . . 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 5096	. . . . 5 ⊢ (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13		0cn 11126	. . . . . . . 8 ⊢ 0 ∈ ℂ
14	13	mulridi 11138	. . . . . . 7 ⊢ (0 · 1) = 0
15	14	opeq2i 4831	. . . . . 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 11037	. . . . . . . 8 ⊢ i = ⟨0R, 1R⟩
21	20	fveq1i 6827	. . . . . . 7 ⊢ (i‘1) = (⟨0R, 1R⟩‘1)
22		df-fv 6494	. . . . . . 7 ⊢ (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
23	21, 22	eqtri 2752	. . . . . 6 ⊢ (i‘1) = (℩𝑦1⟨0R, 1R⟩𝑦)
24	23	breq2i 5103	. . . . 5 ⊢ (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R⟩𝑦))
25		df-r 11038	. . . . . . 7 ⊢ ℝ = (R × {0R})
26		sseq2 3964	. . . . . . . . 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 4555	. . . . . . . 8 ⊢ 𝒫 ℝ = {𝑧 ∣ 𝑧 ⊆ ℝ}
29		df-pw 4555	. . . . . . . 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 5101	. . . . 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 3438   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4553  {csn 4579  ⟨cop 4585   class class class wbr 5095   × cxp 5621  ℩cio 6440  ‘cfv 6486  (class class class)co 7353  Rcnr 10778  0_Rc0r 10779  1_Rc1r 10780  ℝcr 11027  0cc0 11028  1c1 11029  ici 11030   · cmul 11033

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 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-mulcl 11090  ax-mulcom 11092  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1rid 11098  ax-cnre 11101

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356  df-i 11037  df-r 11038

This theorem is referenced by: (None)