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Theorem avril1 20797
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.) (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  |-  -.  ( A ~P RR ( _i
`  1 )  /\  F (/) ( 0  x.  1 ) )

Proof of Theorem avril1
StepHypRef Expression
1 equid 1818 . . . . . . . 8  |-  x  =  x
2 dfnul2 3432 . . . . . . . . . 10  |-  (/)  =  {
x  |  -.  x  =  x }
32abeq2i 2365 . . . . . . . . 9  |-  ( x  e.  (/)  <->  -.  x  =  x )
43con2bii 324 . . . . . . . 8  |-  ( x  =  x  <->  -.  x  e.  (/) )
51, 4mpbi 201 . . . . . . 7  |-  -.  x  e.  (/)
6 eleq1 2318 . . . . . . 7  |-  ( x  =  <. F ,  0
>.  ->  ( x  e.  (/) 
<-> 
<. F ,  0 >.  e.  (/) ) )
75, 6mtbii 295 . . . . . 6  |-  ( x  =  <. F ,  0
>.  ->  -.  <. F , 
0 >.  e.  (/) )
87vtocleg 2829 . . . . 5  |-  ( <. F ,  0 >.  e. 
_V  ->  -.  <. F , 
0 >.  e.  (/) )
9 elex 2771 . . . . . 6  |-  ( <. F ,  0 >.  e.  (/)  ->  <. F ,  0
>.  e.  _V )
109con3i 129 . . . . 5  |-  ( -. 
<. F ,  0 >.  e.  _V  ->  -.  <. F , 
0 >.  e.  (/) )
118, 10pm2.61i 158 . . . 4  |-  -.  <. F ,  0 >.  e.  (/)
12 df-br 3998 . . . . 5  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
( 0  x.  1 ) >.  e.  (/) )
13 0cn 8799 . . . . . . . 8  |-  0  e.  CC
1413mulid1i 8807 . . . . . . 7  |-  ( 0  x.  1 )  =  0
1514opeq2i 3774 . . . . . 6  |-  <. F , 
( 0  x.  1 ) >.  =  <. F ,  0 >.
1615eleq1i 2321 . . . . 5  |-  ( <. F ,  ( 0  x.  1 ) >.  e.  (/)  <->  <. F ,  0
>.  e.  (/) )
1712, 16bitri 242 . . . 4  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
0 >.  e.  (/) )
1811, 17mtbir 292 . . 3  |-  -.  F (/) ( 0  x.  1 )
1918intnan 885 . 2  |-  -.  ( A ~P ( R.  X.  { 0R } ) U. { y  |  (
<. 0R ,  1R >. " { 1 } )  =  { y } }  /\  F (/) ( 0  x.  1 ) )
20 df-i 8714 . . . . . . . 8  |-  _i  =  <. 0R ,  1R >.
2120fveq1i 5459 . . . . . . 7  |-  ( _i
`  1 )  =  ( <. 0R ,  1R >. `  1 )
22 df-fv 4689 . . . . . . 7  |-  ( <. 0R ,  1R >. `  1
)  =  U. {
y  |  ( <. 0R ,  1R >. " {
1 } )  =  { y } }
2321, 22eqtri 2278 . . . . . 6  |-  ( _i
`  1 )  = 
U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } }
2423breq2i 4005 . . . . 5  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P RR U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } } )
25 df-r 8715 . . . . . . 7  |-  RR  =  ( R.  X.  { 0R } )
26 sseq2 3175 . . . . . . . . 9  |-  ( RR  =  ( R.  X.  { 0R } )  -> 
( z  C_  RR  <->  z 
C_  ( R.  X.  { 0R } ) ) )
2726abbidv 2372 . . . . . . . 8  |-  ( RR  =  ( R.  X.  { 0R } )  ->  { z  |  z 
C_  RR }  =  { z  |  z 
C_  ( R.  X.  { 0R } ) } )
28 df-pw 3601 . . . . . . . 8  |-  ~P RR  =  { z  |  z 
C_  RR }
29 df-pw 3601 . . . . . . . 8  |-  ~P ( R.  X.  { 0R }
)  =  { z  |  z  C_  ( R.  X.  { 0R }
) }
3027, 28, 293eqtr4g 2315 . . . . . . 7  |-  ( RR  =  ( R.  X.  { 0R } )  ->  ~P RR  =  ~P ( R.  X.  { 0R }
) )
3125, 30ax-mp 10 . . . . . 6  |-  ~P RR  =  ~P ( R.  X.  { 0R } )
3231breqi 4003 . . . . 5  |-  ( A ~P RR U. {
y  |  ( <. 0R ,  1R >. " {
1 } )  =  { y } }  <->  A ~P ( R.  X.  { 0R } ) U. { y  |  (
<. 0R ,  1R >. " { 1 } )  =  { y } } )
3324, 32bitri 242 . . . 4  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P ( R.  X.  { 0R } ) U. {
y  |  ( <. 0R ,  1R >. " {
1 } )  =  { y } }
)
3433anbi1i 679 . . 3  |-  ( ( A ~P RR ( _i `  1 )  /\  F (/) ( 0  x.  1 ) )  <-> 
( A ~P ( R.  X.  { 0R }
) U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } }  /\  F (/) ( 0  x.  1 ) ) )
3534notbii 289 . 2  |-  ( -.  ( A ~P RR ( _i `  1 )  /\  F (/) ( 0  x.  1 ) )  <->  -.  ( A ~P ( R.  X.  { 0R }
) U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } }  /\  F (/) ( 0  x.  1 ) ) )
3619, 35mpbir 202 1  |-  -.  ( A ~P RR ( _i
`  1 )  /\  F (/) ( 0  x.  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2244   _Vcvv 2763    C_ wss 3127   (/)c0 3430   ~Pcpw 3599   {csn 3614   <.cop 3617   U.cuni 3801   class class class wbr 3997    X. cxp 4659   "cima 4664   ` cfv 4673  (class class class)co 5792   R.cnr 8457   0Rc0r 8458   1Rc1r 8459   RRcr 8704   0cc0 8705   1c1 8706   _ici 8707    x. cmul 8710
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-mulcl 8767  ax-mulcom 8769  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1rid 8775  ax-cnre 8778
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795  df-i 8714  df-r 8715
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