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Theorem avril1 25043
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 1777 . . . . . . . 8  |-  x  =  x
2 dfnul2 3772 . . . . . . . . . 10  |-  (/)  =  {
x  |  -.  x  =  x }
32abeq2i 2570 . . . . . . . . 9  |-  ( x  e.  (/)  <->  -.  x  =  x )
43con2bii 332 . . . . . . . 8  |-  ( x  =  x  <->  -.  x  e.  (/) )
51, 4mpbi 208 . . . . . . 7  |-  -.  x  e.  (/)
6 eleq1 2515 . . . . . . 7  |-  ( x  =  <. F ,  0
>.  ->  ( x  e.  (/) 
<-> 
<. F ,  0 >.  e.  (/) ) )
75, 6mtbii 302 . . . . . 6  |-  ( x  =  <. F ,  0
>.  ->  -.  <. F , 
0 >.  e.  (/) )
87vtocleg 3166 . . . . 5  |-  ( <. F ,  0 >.  e. 
_V  ->  -.  <. F , 
0 >.  e.  (/) )
9 elex 3104 . . . . . 6  |-  ( <. F ,  0 >.  e.  (/)  ->  <. F ,  0
>.  e.  _V )
109con3i 135 . . . . 5  |-  ( -. 
<. F ,  0 >.  e.  _V  ->  -.  <. F , 
0 >.  e.  (/) )
118, 10pm2.61i 164 . . . 4  |-  -.  <. F ,  0 >.  e.  (/)
12 df-br 4438 . . . . 5  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
( 0  x.  1 ) >.  e.  (/) )
13 0cn 9591 . . . . . . . 8  |-  0  e.  CC
1413mulid1i 9601 . . . . . . 7  |-  ( 0  x.  1 )  =  0
1514opeq2i 4206 . . . . . 6  |-  <. F , 
( 0  x.  1 ) >.  =  <. F ,  0 >.
1615eleq1i 2520 . . . . 5  |-  ( <. F ,  ( 0  x.  1 ) >.  e.  (/)  <->  <. F ,  0
>.  e.  (/) )
1712, 16bitri 249 . . . 4  |-  ( F
(/) ( 0  x.  1 )  <->  <. F , 
0 >.  e.  (/) )
1811, 17mtbir 299 . . 3  |-  -.  F (/) ( 0  x.  1 )
1918intnan 914 . 2  |-  -.  ( A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y
)  /\  F (/) ( 0  x.  1 ) )
20 df-i 9504 . . . . . . . 8  |-  _i  =  <. 0R ,  1R >.
2120fveq1i 5857 . . . . . . 7  |-  ( _i
`  1 )  =  ( <. 0R ,  1R >. `  1 )
22 df-fv 5586 . . . . . . 7  |-  ( <. 0R ,  1R >. `  1
)  =  ( iota y 1 <. 0R ,  1R >. y )
2321, 22eqtri 2472 . . . . . 6  |-  ( _i
`  1 )  =  ( iota y 1
<. 0R ,  1R >. y )
2423breq2i 4445 . . . . 5  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P RR ( iota y 1
<. 0R ,  1R >. y ) )
25 df-r 9505 . . . . . . 7  |-  RR  =  ( R.  X.  { 0R } )
26 sseq2 3511 . . . . . . . . 9  |-  ( RR  =  ( R.  X.  { 0R } )  -> 
( z  C_  RR  <->  z 
C_  ( R.  X.  { 0R } ) ) )
2726abbidv 2579 . . . . . . . 8  |-  ( RR  =  ( R.  X.  { 0R } )  ->  { z  |  z 
C_  RR }  =  { z  |  z 
C_  ( R.  X.  { 0R } ) } )
28 df-pw 3999 . . . . . . . 8  |-  ~P RR  =  { z  |  z 
C_  RR }
29 df-pw 3999 . . . . . . . 8  |-  ~P ( R.  X.  { 0R }
)  =  { z  |  z  C_  ( R.  X.  { 0R }
) }
3027, 28, 293eqtr4g 2509 . . . . . . 7  |-  ( RR  =  ( R.  X.  { 0R } )  ->  ~P RR  =  ~P ( R.  X.  { 0R }
) )
3125, 30ax-mp 5 . . . . . 6  |-  ~P RR  =  ~P ( R.  X.  { 0R } )
3231breqi 4443 . . . . 5  |-  ( A ~P RR ( iota y 1 <. 0R ,  1R >. y )  <->  A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y ) )
3324, 32bitri 249 . . . 4  |-  ( A ~P RR ( _i
`  1 )  <->  A ~P ( R.  X.  { 0R } ) ( iota y 1 <. 0R ,  1R >. y ) )
3433anbi1i 695 . . 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 296 . 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 209 1  |-  -.  ( A ~P RR ( _i
`  1 )  /\  F (/) ( 0  x.  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428   _Vcvv 3095    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   {csn 4014   <.cop 4020   class class class wbr 4437    X. cxp 4987   iotacio 5539   ` cfv 5578  (class class class)co 6281   R.cnr 9246   0Rc0r 9247   1Rc1r 9248   RRcr 9494   0cc0 9495   1c1 9496   _ici 9497    x. cmul 9500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-mulcl 9557  ax-mulcom 9559  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1rid 9565  ax-cnre 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-i 9504  df-r 9505
This theorem is referenced by: (None)
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