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

Theorem avril1 26450
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 1889 . . . . . . . 8 𝑥 = 𝑥
2 dfnul2 3779 . . . . . . . . . 10 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
32abeq2i 2626 . . . . . . . . 9 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
43con2bii 345 . . . . . . . 8 (𝑥 = 𝑥 ↔ ¬ 𝑥 ∈ ∅)
51, 4mpbi 218 . . . . . . 7 ¬ 𝑥 ∈ ∅
6 eleq1 2580 . . . . . . 7 (𝑥 = ⟨𝐹, 0⟩ → (𝑥 ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅))
75, 6mtbii 314 . . . . . 6 (𝑥 = ⟨𝐹, 0⟩ → ¬ ⟨𝐹, 0⟩ ∈ ∅)
87vtocleg 3156 . . . . 5 (⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
9 elex 3089 . . . . . 6 (⟨𝐹, 0⟩ ∈ ∅ → ⟨𝐹, 0⟩ ∈ V)
109con3i 148 . . . . 5 (¬ ⟨𝐹, 0⟩ ∈ V → ¬ ⟨𝐹, 0⟩ ∈ ∅)
118, 10pm2.61i 174 . . . 4 ¬ ⟨𝐹, 0⟩ ∈ ∅
12 df-br 4482 . . . . 5 (𝐹∅(0 · 1) ↔ ⟨𝐹, (0 · 1)⟩ ∈ ∅)
13 0cn 9786 . . . . . . . 8 0 ∈ ℂ
1413mulid1i 9796 . . . . . . 7 (0 · 1) = 0
1514opeq2i 4242 . . . . . 6 𝐹, (0 · 1)⟩ = ⟨𝐹, 0⟩
1615eleq1i 2583 . . . . 5 (⟨𝐹, (0 · 1)⟩ ∈ ∅ ↔ ⟨𝐹, 0⟩ ∈ ∅)
1712, 16bitri 262 . . . 4 (𝐹∅(0 · 1) ↔ ⟨𝐹, 0⟩ ∈ ∅)
1811, 17mtbir 311 . . 3 ¬ 𝐹∅(0 · 1)
1918intnan 950 . 2 ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1))
20 df-i 9699 . . . . . . . 8 i = ⟨0R, 1R
2120fveq1i 5987 . . . . . . 7 (i‘1) = (⟨0R, 1R⟩‘1)
22 df-fv 5697 . . . . . . 7 (⟨0R, 1R⟩‘1) = (℩𝑦1⟨0R, 1R𝑦)
2321, 22eqtri 2536 . . . . . 6 (i‘1) = (℩𝑦1⟨0R, 1R𝑦)
2423breq2i 4489 . . . . 5 (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R𝑦))
25 df-r 9700 . . . . . . 7 ℝ = (R × {0R})
26 sseq2 3494 . . . . . . . . 9 (ℝ = (R × {0R}) → (𝑧 ⊆ ℝ ↔ 𝑧 ⊆ (R × {0R})))
2726abbidv 2632 . . . . . . . 8 (ℝ = (R × {0R}) → {𝑧𝑧 ⊆ ℝ} = {𝑧𝑧 ⊆ (R × {0R})})
28 df-pw 4013 . . . . . . . 8 𝒫 ℝ = {𝑧𝑧 ⊆ ℝ}
29 df-pw 4013 . . . . . . . 8 𝒫 (R × {0R}) = {𝑧𝑧 ⊆ (R × {0R})}
3027, 28, 293eqtr4g 2573 . . . . . . 7 (ℝ = (R × {0R}) → 𝒫 ℝ = 𝒫 (R × {0R}))
3125, 30ax-mp 5 . . . . . 6 𝒫 ℝ = 𝒫 (R × {0R})
3231breqi 4487 . . . . 5 (𝐴𝒫 ℝ(℩𝑦1⟨0R, 1R𝑦) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦))
3324, 32bitri 262 . . . 4 (𝐴𝒫 ℝ(i‘1) ↔ 𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦))
3433anbi1i 726 . . 3 ((𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1)))
3534notbii 308 . 2 (¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1)) ↔ ¬ (𝐴𝒫 (R × {0R})(℩𝑦1⟨0R, 1R𝑦) ∧ 𝐹∅(0 · 1)))
3619, 35mpbir 219 1 ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wcel 1938  {cab 2500  Vcvv 3077  wss 3444  c0 3777  𝒫 cpw 4011  {csn 4028  cop 4034   class class class wbr 4481   × cxp 4930  cio 5651  cfv 5689  (class class class)co 6425  Rcnr 9441  0Rc0r 9442  1Rc1r 9443  cr 9689  0cc0 9690  1c1 9691  ici 9692   · cmul 9695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-resscn 9747  ax-1cn 9748  ax-icn 9749  ax-addcl 9750  ax-mulcl 9752  ax-mulcom 9754  ax-mulass 9756  ax-distr 9757  ax-i2m1 9758  ax-1rid 9760  ax-cnre 9763
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697  df-ov 6428  df-i 9699  df-r 9700
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator