Theorem avril1 9937

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 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.)

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	|- -. (A~PRR(_i` 1) /\ F(/)(0 x. 1))

Proof of Theorem avril1

Step	Hyp	Ref	Expression
1		equid 1322	. . . . . . . 8 |- x = x
2		dfnul2 2703	. . . . . . . . . 10 |- (/) = {x | -. x = x}
3	2	abeq2i 1838	. . . . . . . . 9 |- (x e. (/) <-> -. x = x)
4	3	con2bii 237	. . . . . . . 8 |- (x = x <-> -. x e. (/))
5	1, 4	mpbi 205	. . . . . . 7 |- -. x e. (/)
6		eleq1 1794	. . . . . . 7 |- (x = <.F, 0>. -> (x e. (/) <-> <.F, 0>. e. (/)))
7	5, 6	mtbii 781	. . . . . 6 |- (x = <.F, 0>. -> -. <.F, 0>. e. (/))
8	7	vtocleg 2188	. . . . 5 |- (<.F, 0>. e. _V -> -. <.F, 0>. e. (/))
9		elisset 2132	. . . . . 6 |- (<.F, 0>. e. (/) -> <.F, 0>. e. _V)
10	9	con3i 113	. . . . 5 |- (-. <.F, 0>. e. _V -> -. <.F, 0>. e. (/))
11	8, 10	pm2.61i 139	. . . 4 |- -. <.F, 0>. e. (/)
12		df-br 3159	. . . . 5 |- (F(/)(0 x. 1) <-> <.F, (0 x. 1)>. e. (/))
13		0cn 6277	. . . . . . . 8 |- 0 e. CC
14	13	mulid1i 6281	. . . . . . 7 |- (0 x. 1) = 0
15	14	opeq2i 2984	. . . . . 6 |- <.F, (0 x. 1)>. = <.F, 0>.
16	15	eleq1i 1797	. . . . 5 |- (<.F, (0 x. 1)>. e. (/) <-> <.F, 0>. e. (/))
17	12, 16	bitri 189	. . . 4 |- (F(/)(0 x. 1) <-> <.F, 0>. e. (/))
18	11, 17	mtbir 208	. . 3 |- -. F(/)(0 x. 1)
19	18	intnan 752	. 2 |- -. (A~P(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1))
20		df-i 6191	. . . . . . . 8 |- _i = <.0R, 1R>.
21	20	fveq1i 4493	. . . . . . 7 |- (_i` 1) = (<.0R, 1R>.` 1)
22		df-fv 3825	. . . . . . 7 |- (<.0R, 1R>.` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
23	21, 22	eqtri 1745	. . . . . 6 |- (_i` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
24	23	breq2i 3166	. . . . 5 |- (A~PRR(_i` 1) <-> A~PRRU.{y | (<.0R, 1R>."{1}) = {y}})
25		df-r 6192	. . . . . . 7 |- RR = (R. X. {0R})
26		sseq2 2472	. . . . . . . . 9 |- (RR = (R. X. {0R}) -> (z C_ RR <-> z C_ (R. X. {0R})))
27	26	abbidv 1845	. . . . . . . 8 |- (RR = (R. X. {0R}) -> {z | z C_ RR} = {z | z C_ (R. X. {0R})})
28		df-pw 2859	. . . . . . . 8 |- ~PRR = {z | z C_ RR}
29		df-pw 2859	. . . . . . . 8 |- ~P(R. X. {0R}) = {z | z C_ (R. X. {0R})}
30	27, 28, 29	3eqtr4g 1790	. . . . . . 7 |- (RR = (R. X. {0R}) -> ~PRR = ~P(R. X. {0R}))
31	25, 30	ax-mp 7	. . . . . 6 |- ~PRR = ~P(R. X. {0R})
32	31	breqi 3164	. . . . 5 |- (A~PRRU.{y | (<.0R, 1R>."{1}) = {y}} <-> A~P(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
33	24, 32	bitri 189	. . . 4 |- (A~PRR(_i` 1) <-> A~P(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
34	33	anbi1i 536	. . 3 |- ((A~PRR(_i` 1) /\ F(/)(0 x. 1)) <-> (A~P(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1)))
35	34	notbii 203	. 2 |- (-. (A~PRR(_i` 1) /\ F(/)(0 x. 1)) <-> -. (A~P(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1)))
36	19, 35	mpbir 206	1 |- -. (A~PRR(_i` 1) /\ F(/)(0 x. 1))

Syntax hints:  -. wn 2   /\ wa 239   = wceq 1136   e. wcel 1138  {cab 1708  _Vcvv 2125   C_ wss 2426  (/)c0 2701  ~Pcpw 2856  {csn 2868  <.cop 2870  U.cuni 2999   class class class wbr 3158   X. cxp 3795  "cima 3800  ` cfv 3809  (class class class)co 4695  R.cnr 5941  0Rc0r 5942  1Rc1r 5943  RRcr 6181  0cc0 6182  1c1 6183  _ici 6184   x. cmul 6187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-rep 3243  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601  ax-inf2 5540

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-ral 1943  df-rex 1944  df-reu 1945  df-rab 1946  df-v 2127  df-sbc 2287  df-csb 2374  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-int 3037  df-iun 3079  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-id 3401  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761  df-xp 3811  df-rel 3812  df-cnv 3813  df-co 3814  df-dm 3815  df-rn 3816  df-res 3817  df-ima 3818  df-fun 3819  df-fn 3820  df-f 3821  df-fv 3825  df-opr 4697  df-oprab 4698  df-1st 4831  df-2nd 4832  df-rdg 4951  df-1o 4988  df-oadd 4990  df-omul 4991  df-er 5129  df-ec 5131  df-qs 5134  df-ni 5948  df-pli 5949  df-mi 5950  df-lti 5951  df-plpq 5983  df-mpq 5984  df-enq 5985  df-nq 5986  df-plq 5987  df-mq 5988  df-rq 5989  df-ltq 5990  df-1q 5991  df-np 6034  df-1p 6035  df-plp 6036  df-mp 6037  df-ltp 6038  df-plpr 6112  df-mpr 6113  df-enr 6114  df-nr 6115  df-plr 6116  df-mr 6117  df-0r 6119  df-1r 6120  df-m1r 6121  df-c 6188  df-0 6189  df-1 6190  df-i 6191  df-r 6192  df-plus 6193  df-mul 6194

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