Theorem avril1 8779

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

Proof of Theorem avril1

Step	Hyp	Ref	Expression
1		equid 1128	. . . . . . . 8 |- x = x
2		dfnul2 2285	. . . . . . . . . 10 |- (/) = {x | -. x = x}
3	2	abeq2i 1573	. . . . . . . . 9 |- (x e. (/) <-> -. x = x)
4	3	con2bii 221	. . . . . . . 8 |- (x = x <-> -. x e. (/))
5	1, 4	mpbi 189	. . . . . . 7 |- -. x e. (/)
6		eleq1 1537	. . . . . . 7 |- (x = <.F, 0>. -> (x e. (/) <-> <.F, 0>. e. (/)))
7	5, 6	mtbii 718	. . . . . 6 |- (x = <.F, 0>. -> -. <.F, 0>. e. (/))
8	7	vtocleg 1858	. . . . 5 |- (<.F, 0>. e. V -> -. <.F, 0>. e. (/))
9		elisset 1820	. . . . . 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 2625	. . . . 5 |- (F(/)(0 x. 1) <-> <.F, (0 x. 1)>. e. (/))
13		0cn 5340	. . . . . . . 8 |- 0 e. CC
14	13	mulid1 5344	. . . . . . 7 |- (0 x. 1) = 0
15	14	opeq2i 2495	. . . . . 6 |- <.F, (0 x. 1)>. = <.F, 0>.
16	15	eleq1i 1540	. . . . 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 693	. 2 |- -. (AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1))
20		df-i 5255	. . . . . . . 8 |- i = <.0R, 1R>.
21	20	fveq1i 3731	. . . . . . 7 |- (i` 1) = (<.0R, 1R>.` 1)
22		df-fv 3204	. . . . . . 7 |- (<.0R, 1R>.` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
23	21, 22	eqtr 1498	. . . . . 6 |- (i` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
24	23	breq2i 2632	. . . . 5 |- (AP~RR(i` 1) <-> AP~RRU.{y | (<.0R, 1R>."{1}) = {y}})
25		df-r 5256	. . . . . . 7 |- RR = (R. X. {0R})
26		sseq2 2086	. . . . . . . . 9 |- (RR = (R. X. {0R}) -> (z (_ RR <-> z (_ (R. X. {0R})))
27	26	abbidv 1580	. . . . . . . 8 |- (RR = (R. X. {0R}) -> {z | z (_ RR} = {z | z (_ (R. X. {0R})})
28		df-pw 2406	. . . . . . . 8 |- P~RR = {z | z (_ RR}
29		df-pw 2406	. . . . . . . 8 |- P~(R. X. {0R}) = {z | z (_ (R. X. {0R})}
30	27, 28, 29	3eqtr4g 1534	. . . . . . 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 2630	. . . . 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 483	. . 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 958   e. wcel 960  {cab 1466  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  {csn 2413  <.cop 2415  U.cuni 2507   class class class wbr 2624   X. cxp 3174  "cima 3179  ` cfv 3188  (class class class)co 3969  R.cnr 5005  0Rc0r 5006  1Rc1r 5007  RRcr 5245  0cc0 5246  1c1 5247  ici 5248   x. cmul 5251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258

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