Theorem avril1 9058

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 1162	. . . . . . . 8 |- x = x
2		dfnul2 2334	. . . . . . . . . 10 |- (/) = {x | -. x = x}
3	2	abeq2i 1613	. . . . . . . . 9 |- (x e. (/) <-> -. x = x)
4	3	con2bii 219	. . . . . . . 8 |- (x = x <-> -. x e. (/))
5	1, 4	mpbi 187	. . . . . . 7 |- -. x e. (/)
6		eleq1 1577	. . . . . . 7 |- (x = <.F, 0>. -> (x e. (/) <-> <.F, 0>. e. (/)))
7	5, 6	mtbii 721	. . . . . 6 |- (x = <.F, 0>. -> -. <.F, 0>. e. (/))
8	7	vtocleg 1901	. . . . 5 |- (<.F, 0>. e. V -> -. <.F, 0>. e. (/))
9		elisset 1863	. . . . . 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 124	. . . 4 |- -. <.F, 0>. e. (/)
12		df-br 2693	. . . . 5 |- (F(/)(0 x. 1) <-> <.F, (0 x. 1)>. e. (/))
13		0cn 5482	. . . . . . . 8 |- 0 e. CC
14	13	mulid1i 5486	. . . . . . 7 |- (0 x. 1) = 0
15	14	opeq2i 2556	. . . . . 6 |- <.F, (0 x. 1)>. = <.F, 0>.
16	15	eleq1i 1580	. . . . 5 |- (<.F, (0 x. 1)>. e. (/) <-> <.F, 0>. e. (/))
17	12, 16	bitri 171	. . . 4 |- (F(/)(0 x. 1) <-> <.F, 0>. e. (/))
18	11, 17	mtbir 190	. . 3 |- -. F(/)(0 x. 1)
19	18	intnan 695	. 2 |- -. (AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}} /\ F(/)(0 x. 1))
20		df-i 5397	. . . . . . . 8 |- i = <.0R, 1R>.
21	20	fveq1i 3836	. . . . . . 7 |- (i` 1) = (<.0R, 1R>.` 1)
22		df-fv 3279	. . . . . . 7 |- (<.0R, 1R>.` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
23	21, 22	eqtri 1538	. . . . . 6 |- (i` 1) = U.{y | (<.0R, 1R>."{1}) = {y}}
24	23	breq2i 2700	. . . . 5 |- (AP~RR(i` 1) <-> AP~RRU.{y | (<.0R, 1R>."{1}) = {y}})
25		df-r 5398	. . . . . . 7 |- RR = (R. X. {0R})
26		sseq2 2135	. . . . . . . . 9 |- (RR = (R. X. {0R}) -> (z (_ RR <-> z (_ (R. X. {0R})))
27	26	abbidv 1620	. . . . . . . 8 |- (RR = (R. X. {0R}) -> {z | z (_ RR} = {z | z (_ (R. X. {0R})})
28		df-pw 2459	. . . . . . . 8 |- P~RR = {z | z (_ RR}
29		df-pw 2459	. . . . . . . 8 |- P~(R. X. {0R}) = {z | z (_ (R. X. {0R})}
30	27, 28, 29	3eqtr4g 1574	. . . . . . 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 2698	. . . . 5 |- (AP~RRU.{y | (<.0R, 1R>."{1}) = {y}} <-> AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
33	24, 32	bitri 171	. . . 4 |- (AP~RR(i` 1) <-> AP~(R. X. {0R})U.{y | (<.0R, 1R>."{1}) = {y}})
34	33	anbi1i 484	. . 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	notbii 185	. 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 188	1 |- -. (AP~RR(i` 1) /\ F(/)(0 x. 1))

Syntax hints:  -. wn 2   /\ wa 221   = wceq 992   e. wcel 994  {cab 1505  Vcvv 1857   (_ wss 2099  (/)c0 2332  P~cpw 2458  {csn 2467  <.cop 2469  U.cuni 2569   class class class wbr 2692   X. cxp 3249  "cima 3254  ` cfv 3263  (class class class)co 4021  R.cnr 5147  0Rc0r 5148  1Rc1r 5149  RRcr 5387  0cc0 5388  1c1 5389  ici 5390   x. cmul 5393

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400

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