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Theorem avril1 21757
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 german 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/mpeuni/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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 equid 1688 . . . . . . . 8  |-  x  =  x
2 dfnul2 3630 . . . . . . . . . 10  |-  (/)  =  {
x  |  -.  x  =  x }
32abeq2i 2543 . . . . . . . . 9  |-  ( x  e.  (/)  <->  -.  x  =  x )
43con2bii 323 . . . . . . . 8  |-  ( x  =  x  <->  -.  x  e.  (/) )
51, 4mpbi 200 . . . . . . 7  |-  -.  x  e.  (/)
6 eleq1 2496 . . . . . . 7  |-  ( x  =  <. F ,  0
>.  ->  ( x  e.  (/) 
<-> 
<. F ,  0 >.  e.  (/) ) )
75, 6mtbii 294 . . . . . 6  |-  ( x  =  <. F ,  0
>.  ->  -.  <. F , 
0 >.  e.  (/) )
87vtocleg 3022 . . . . 5  |-  ( <. F ,  0 >.  e. 
_V  ->  -.  <. F , 
0 >.  e.  (/) )
9 elex 2964 . . . . . 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 4213 . . . . 5  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
( 0  x.  1 ) >.  e.  (/) )
13 0cn 9084 . . . . . . . 8  |-  0  e.  CC
1413mulid1i 9092 . . . . . . 7  |-  ( 0  x.  1 )  =  0
1514opeq2i 3988 . . . . . 6  |-  <. F , 
( 0  x.  1 ) >.  =  <. F ,  0 >.
1615eleq1i 2499 . . . . 5  |-  ( <. F ,  ( 0  x.  1 ) >.  e.  (/)  <->  <. F ,  0
>.  e.  (/) )
1712, 16bitri 241 . . . 4  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
0 >.  e.  (/) )
1811, 17mtbir 291 . . 3  |-  -.  F (/) ( 0  x.  1 )
1918intnan 881 . 2  |-  -.  ( A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y
)  /\  F (/) ( 0  x.  1 ) )
20 df-i 8999 . . . . . . . 8  |-  _i  =  <. 0R ,  1R >.
2120fveq1i 5729 . . . . . . 7  |-  ( _i
`  1 )  =  ( <. 0R ,  1R >. `  1 )
22 df-fv 5462 . . . . . . 7  |-  ( <. 0R ,  1R >. `  1
)  =  ( iota y 1 <. 0R ,  1R >. y )
2321, 22eqtri 2456 . . . . . 6  |-  ( _i
`  1 )  =  ( iota y 1
<. 0R ,  1R >. y )
2423breq2i 4220 . . . . 5  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P RR ( iota y 1
<. 0R ,  1R >. y ) )
25 df-r 9000 . . . . . . 7  |-  RR  =  ( R.  X.  { 0R } )
26 sseq2 3370 . . . . . . . . 9  |-  ( RR  =  ( R.  X.  { 0R } )  -> 
( z  C_  RR  <->  z 
C_  ( R.  X.  { 0R } ) ) )
2726abbidv 2550 . . . . . . . 8  |-  ( RR  =  ( R.  X.  { 0R } )  ->  { z  |  z 
C_  RR }  =  { z  |  z 
C_  ( R.  X.  { 0R } ) } )
28 df-pw 3801 . . . . . . . 8  |-  ~P RR  =  { z  |  z 
C_  RR }
29 df-pw 3801 . . . . . . . 8  |-  ~P ( R.  X.  { 0R }
)  =  { z  |  z  C_  ( R.  X.  { 0R }
) }
3027, 28, 293eqtr4g 2493 . . . . . . 7  |-  ( RR  =  ( R.  X.  { 0R } )  ->  ~P RR  =  ~P ( R.  X.  { 0R }
) )
3125, 30ax-mp 8 . . . . . 6  |-  ~P RR  =  ~P ( R.  X.  { 0R } )
3231breqi 4218 . . . . 5  |-  ( A ~P RR ( iota y 1 <. 0R ,  1R >. y )  <->  A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y ) )
3324, 32bitri 241 . . . 4  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y ) )
3433anbi1i 677 . . 3  |-  ( ( A ~P RR ( _i `  1 )  /\  F (/) ( 0  x.  1 ) )  <-> 
( A ~P ( R.  X.  { 0R }
) ( iota y
1 <. 0R ,  1R >. y )  /\  F (/) ( 0  x.  1 ) ) )
3534notbii 288 . 2  |-  ( -.  ( A ~P RR ( _i `  1 )  /\  F (/) ( 0  x.  1 ) )  <->  -.  ( A ~P ( R.  X.  { 0R }
) ( iota y
1 <. 0R ,  1R >. y )  /\  F (/) ( 0  x.  1 ) ) )
3619, 35mpbir 201 1  |-  -.  ( A ~P RR ( _i
`  1 )  /\  F (/) ( 0  x.  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   <.cop 3817   class class class wbr 4212    X. cxp 4876   iotacio 5416   ` cfv 5454  (class class class)co 6081   R.cnr 8742   0Rc0r 8743   1Rc1r 8744   RRcr 8989   0cc0 8990   1c1 8991   _ici 8992    x. cmul 8995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-mulcom 9054  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1rid 9060  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-i 8999  df-r 9000
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