MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  avril1 Unicode version

Theorem avril1 20761
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 3399 . . . . . . . . . 10  |-  (/)  =  {
x  |  -.  x  =  x }
32abeq2i 2363 . . . . . . . . 9  |-  ( x  e.  (/)  <->  -.  x  =  x )
43con2bii 324 . . . . . . . 8  |-  ( x  =  x  <->  -.  x  e.  (/) )
51, 4mpbi 201 . . . . . . 7  |-  -.  x  e.  (/)
6 eleq1 2316 . . . . . . 7  |-  ( x  =  <. F ,  0
>.  ->  ( x  e.  (/) 
<-> 
<. F ,  0 >.  e.  (/) ) )
75, 6mtbii 295 . . . . . 6  |-  ( x  =  <. F ,  0
>.  ->  -.  <. F , 
0 >.  e.  (/) )
87vtocleg 2805 . . . . 5  |-  ( <. F ,  0 >.  e. 
_V  ->  -.  <. F , 
0 >.  e.  (/) )
9 elex 2748 . . . . . 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 3964 . . . . 5  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
( 0  x.  1 ) >.  e.  (/) )
13 0cn 8764 . . . . . . . 8  |-  0  e.  CC
1413mulid1i 8772 . . . . . . 7  |-  ( 0  x.  1 )  =  0
1514opeq2i 3741 . . . . . 6  |-  <. F , 
( 0  x.  1 ) >.  =  <. F ,  0 >.
1615eleq1i 2319 . . . . 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 8679 . . . . . . . 8  |-  _i  =  <. 0R ,  1R >.
2120fveq1i 5424 . . . . . . 7  |-  ( _i
`  1 )  =  ( <. 0R ,  1R >. `  1 )
22 df-fv 4654 . . . . . . 7  |-  ( <. 0R ,  1R >. `  1
)  =  U. {
y  |  ( <. 0R ,  1R >. " {
1 } )  =  { y } }
2321, 22eqtri 2276 . . . . . 6  |-  ( _i
`  1 )  = 
U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } }
2423breq2i 3971 . . . . 5  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P RR U. { y  |  ( <. 0R ,  1R >. " { 1 } )  =  { y } } )
25 df-r 8680 . . . . . . 7  |-  RR  =  ( R.  X.  { 0R } )
26 sseq2 3142 . . . . . . . . 9  |-  ( RR  =  ( R.  X.  { 0R } )  -> 
( z  C_  RR  <->  z 
C_  ( R.  X.  { 0R } ) ) )
2726abbidv 2370 . . . . . . . 8  |-  ( RR  =  ( R.  X.  { 0R } )  ->  { z  |  z 
C_  RR }  =  { z  |  z 
C_  ( R.  X.  { 0R } ) } )
28 df-pw 3568 . . . . . . . 8  |-  ~P RR  =  { z  |  z 
C_  RR }
29 df-pw 3568 . . . . . . . 8  |-  ~P ( R.  X.  { 0R }
)  =  { z  |  z  C_  ( R.  X.  { 0R }
) }
3027, 28, 293eqtr4g 2313 . . . . . . 7  |-  ( RR  =  ( R.  X.  { 0R } )  ->  ~P RR  =  ~P ( R.  X.  { 0R }
) )
3125, 30ax-mp 10 . . . . . 6  |-  ~P RR  =  ~P ( R.  X.  { 0R } )
3231breqi 3969 . . . . 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 2242   _Vcvv 2740    C_ wss 3094   (/)c0 3397   ~Pcpw 3566   {csn 3581   <.cop 3584   U.cuni 3768   class class class wbr 3963    X. cxp 4624   "cima 4629   ` cfv 4638  (class class class)co 5757   R.cnr 8422   0Rc0r 8423   1Rc1r 8424   RRcr 8669   0cc0 8670   1c1 8671   _ici 8672    x. cmul 8675
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 2237  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-mulcl 8732  ax-mulcom 8734  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1rid 8740  ax-cnre 8743
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 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fv 4654  df-ov 5760  df-i 8679  df-r 8680
  Copyright terms: Public domain W3C validator