MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  avril1 Structured version   Visualization version   GIF version

Theorem avril1 27449
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.
StepHypRef Expression
1 equid 1985 . . . . . . . 8 𝑥 = 𝑥
2 dfnul2 3950 . . . . . . . . . 10 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
32abeq2i 2764 . . . . . . . . 9 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
43con2bii 346 . . . . . . . 8 (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅)
51, 4mpbi 220 . . . . . . 7 ¬ 𝑥 ∈ ∅
6 eleq1 2718 . . . . . . 7 (𝑥 = ⟨𝐹, 0⟩ → (𝑥 ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅))
75, 6mtbii 315 . . . . . 6 (𝑥 = ⟨𝐹, 0⟩ → ¬ ⟨𝐹, 0⟩ ∈ ∅)
87vtocleg 3310 . . . . 5 (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9 elex 3243 . . . . . 6 (⟨𝐹, 0⟩ ∈ ∅ → ⟨𝐹, 0⟩ ∈ V)
109con3i 150 . . . . 5 (¬ ⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
118, 10pm2.61i 176 . . . 4 ¬ ⟨𝐹, 0⟩ ∈ ∅
12 df-br 4686 . . . . 5 (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13 0cn 10070 . . . . . . . 8 0 ∈ ℂ
1413mulid1i 10080 . . . . . . 7 (0 · 1) = 0
1514opeq2i 4437 . . . . . 6 𝐹, (0 · 1)⟩ = ⟨𝐹, 0⟩
1615eleq1i 2721 . . . . 5 (⟨𝐹, (0 · 1)⟩ ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅)
1712, 16bitri 264 . . . 4 (𝐹∅(0 · 1) ↔ ⟨𝐹, 0⟩ ∈ ∅)
1811, 17mtbir 312 . . 3 ¬ 𝐹∅(0 · 1)
1918intnan 980 . 2 ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1))
20 df-i 9983 . . . . . . . 8 i = ⟨0R, 1R
2120fveq1i 6230 . . . . . . 7 (i‘1) = (⟨0R, 1R⟩‘1)
22 df-fv 5934 . . . . . . 7 (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R𝑦)
2321, 22eqtri 2673 . . . . . 6 (i‘1) = (℩𝑦1⟨0R, 1R𝑦)
2423breq2i 4693 . . . . 5 (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R𝑦))
25 df-r 9984 . . . . . . 7 ℝ = (R × {0R})
26 sseq2 3660 . . . . . . . . 9 (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R})))
2726abbidv 2770 . . . . . . . 8 (ℝ = (R × {0R}) → {𝑧𝑧 ⊆ ℝ} = {𝑧𝑧 ⊆ (R × {0R})})
28 df-pw 4193 . . . . . . . 8 𝒫 ℝ = {𝑧𝑧 ⊆ ℝ}
29 df-pw 4193 . . . . . . . 8 𝒫 (R × {0R}) = {𝑧𝑧 ⊆ (R × {0R})}
3027, 28, 293eqtr4g 2710 . . . . . . 7 (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R}))
3125, 30ax-mp 5 . . . . . 6 𝒫 ℝ = 𝒫 (R × {0R})
3231breqi 4691 . . . . 5 (𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R𝑦) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦))
3324, 32bitri 264 . . . 4 (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦))
3433anbi1i 731 . . 3 ((𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1)))
3534notbii 309 . 2 (¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1)))
3619, 35mpbir 221 1 ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cop 4216   class class class wbr 4685   × cxp 5141  cio 5887  cfv 5926  (class class class)co 6690  Rcnr 9725  0Rc0r 9726  1Rc1r 9727  cr 9973  0cc0 9974  1c1 9975  ici 9976   · cmul 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-mulcl 10036  ax-mulcom 10038  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1rid 10044  ax-cnre 10047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-i 9983  df-r 9984
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator