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Theorem avril1 20829
 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 germanus dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object 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 http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-2006 entry.
Assertion
Ref Expression
avril1

Proof of Theorem avril1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 equid 1644 . . . . . . . 8
2 dfnul2 3457 . . . . . . . . . 10
32abeq2i 2390 . . . . . . . . 9
43con2bii 322 . . . . . . . 8
51, 4mpbi 199 . . . . . . 7
6 eleq1 2343 . . . . . . 7
75, 6mtbii 293 . . . . . 6
87vtocleg 2854 . . . . 5
9 elex 2796 . . . . . 6
109con3i 127 . . . . 5
118, 10pm2.61i 156 . . . 4
12 df-br 4024 . . . . 5
13 0cn 8826 . . . . . . . 8
1413mulid1i 8834 . . . . . . 7
1514opeq2i 3800 . . . . . 6
1615eleq1i 2346 . . . . 5
1712, 16bitri 240 . . . 4
1811, 17mtbir 290 . . 3
1918intnan 880 . 2
20 df-i 8741 . . . . . . . 8
2120fveq1i 5486 . . . . . . 7
22 df-fv 5228 . . . . . . 7
2321, 22eqtri 2303 . . . . . 6
2423breq2i 4031 . . . . 5
25 df-r 8742 . . . . . . 7
26 sseq2 3200 . . . . . . . . 9
2726abbidv 2397 . . . . . . . 8
28 df-pw 3627 . . . . . . . 8
29 df-pw 3627 . . . . . . . 8
3027, 28, 293eqtr4g 2340 . . . . . . 7
3125, 30ax-mp 8 . . . . . 6
3231breqi 4029 . . . . 5
3324, 32bitri 240 . . . 4
3433anbi1i 676 . . 3
3534notbii 287 . 2
3619, 35mpbir 200 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 358   wceq 1623   wcel 1684  cab 2269  cvv 2788   wss 3152  c0 3455  cpw 3625  csn 3640  cop 3643  cuni 3827   class class class wbr 4023   cxp 4685  cima 4690  cfv 5220  (class class class)co 5819  cnr 8484  c0r 8485  c1r 8486  cr 8731  cc0 8732  c1 8733  ci 8734   cmul 8737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-mulcl 8794  ax-mulcom 8796  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1rid 8802  ax-cnre 8805 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4693  df-cnv 4695  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fv 5228  df-ov 5822  df-i 8741  df-r 8742
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