Theorem avril1 8723

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-06 entry.

Assertion

Ref	Expression
avril1	|- -. (AP~RR(i` 1) /\ F(/)(0 x. 1))

Proof of Theorem avril1

Step	Hyp	Ref	Expression
1		equid 1122	. . . . . . . 8 |- x = x
2		dfnul2 2272	. . . . . . . . . 10 |- (/) = {x | -. x = x}
3	2	abeq2i 1562	. . . . . . . . 9 |- (x e. (/) <-> -. x = x)
4	3	con2bii 221	. . . . . . . 8 |- (x = x <-> -. x e. (/))
5	1, 4	mpbi 189	. . . . . . 7 |- -. x e. (/)
6		eleq1 1526	. . . . . . 7 |- (x = <.F, 0>. -> (x e. (/) <-> <.F, 0>. e. (/)))
7	5, 6	mtbii 714	. . . . . 6 |- (x = <.F, 0>. -> -. <.F, 0>. e. (/))
8	7	vtocleg 1846	. . . . 5 |- (<.F, 0>. e. V -> -. <.F, 0>. e. (/))
9		elisset 1808	. . . . . 6 |- (<.F, 0>. e. (/) -> <.F, 0>. e. V)
10	9	con3i 98	. . . . 5 |- (-. <.F, 0>. e. V -> -. <.F, 0>. e. (/))
11	8, 10	pm2.61i 126	. . . 4 |- -. <.F, 0>. e. (/)
12		df-br 2610	. . . . 5 |- (F(/)(0 x. 1) <-> <.F, (0 x. 1)>. e. (/))
13		0cn 5300	. . . . . . . 8 |- 0 e. CC
14	13	mulid1 5304	. . . . . . 7 |- (0 x. 1) = 0
15	14	opeq2i 2482	. . . . . 6 |- <.F, (0 x. 1)>. = <.F, 0>.
16	15	eleq1i 1529	. . . . 5 |- (<.F, (0 x. 1)>. e. (/) <-> <.F, 0>. e. (/))
17	12, 16	bitr 173	. . . 4 |- (F(/)(0 x. 1) <-> <.F, 0>. e. (/))
18	11, 17	mtbir 192	. . 3 |- -. F(/)(0 x. 1)
19	18	intnan 689	. 2 |- -. (AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1))
20		df-i 5215	. . . . . . . 8 |- i = <.0R, 1R>.
21	20	fveq1i 3710	. . . . . . 7 |- (i` 1) = (<.0R, 1R>.` 1)
22		df-fv 3188	. . . . . . 7 |- (<.0R, 1R>.` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
23	21, 22	eqtr 1487	. . . . . 6 |- (i` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
24	23	breq2i 2617	. . . . 5 |- (AP~RR(i` 1) <-> AP~RRU.{y | (<.0R, 1R>."{1}) = {y}})
25		df-r 5216	. . . . . . 7 |- RR = (R. X. {0R})
26		sseq2 2073	. . . . . . . . 9 |- (RR = (R. X. {0R}) -> (z (_ RR <-> z (_ (R. X. {0R})))
27	26	abbidv 1569	. . . . . . . 8 |- (RR = (R. X. {0R}) -> {z | z (_ RR} = {z | z (_ (R. X. {0R})})
28		df-pw 2392	. . . . . . . 8 |- P~RR = {z | z (_ RR}
29		df-pw 2392	. . . . . . . 8 |- P~(R. X. {0R}) = {z | z (_ (R. X. {0R})}
30	27, 28, 29	3eqtr4g 1523	. . . . . . 7 |- (RR = (R. X. {0R}) -> P~RR = P~(R. X. {0R}))
31	25, 30	ax-mp 7	. . . . . 6 |- P~RR = P~(R. X. {0R})
32	31	breqi 2615	. . . . 5 |- (AP~RRU.{y | (<.0R, 1R>."{1}) = {y}} <-> AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
33	24, 32	bitr 173	. . . 4 |- (AP~RR(i` 1) <-> AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
34	33	anbi1i 480	. . 3 |- ((AP~RR(i` 1) /\ F(/)(0 x. 1)) <-> (AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1)))
35	34	negbii 187	. 2 |- (-. (AP~RR(i` 1) /\ F(/)(0 x. 1)) <-> -. (AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1)))
36	19, 35	mpbir 190	1 |- -. (AP~RR(i` 1) /\ F(/)(0 x. 1))

Syntax hints:  -. wn 2   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802   (_ wss 2037  (/)c0 2270  P~cpw 2391  {csn 2399  <.cop 2401  U.cuni 2493   class class class wbr 2609   X. cxp 3158  "cima 3163  ` cfv 3172  (class class class)co 3948  R.cnr 4965  0Rc0r 4966  1Rc1r 4967  RRcr 5205  0cc0 5206  1c1 5207  ici 5208   x. cmul 5211

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218

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