ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  endisj Unicode version

Theorem endisj 6724
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
Hypotheses
Ref Expression
endisj.1  |-  A  e. 
_V
endisj.2  |-  B  e. 
_V
Assertion
Ref Expression
endisj  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem endisj
StepHypRef Expression
1 endisj.1 . . . 4  |-  A  e. 
_V
2 0ex 4061 . . . 4  |-  (/)  e.  _V
31, 2xpsnen 6721 . . 3  |-  ( A  X.  { (/) } ) 
~~  A
4 endisj.2 . . . 4  |-  B  e. 
_V
5 1on 6326 . . . . 5  |-  1o  e.  On
65elexi 2701 . . . 4  |-  1o  e.  _V
74, 6xpsnen 6721 . . 3  |-  ( B  X.  { 1o }
)  ~~  B
83, 7pm3.2i 270 . 2  |-  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)
9 xp01disj 6336 . 2  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
10 p0ex 4118 . . . 4  |-  { (/) }  e.  _V
111, 10xpex 4660 . . 3  |-  ( A  X.  { (/) } )  e.  _V
126snex 4115 . . . 4  |-  { 1o }  e.  _V
134, 12xpex 4660 . . 3  |-  ( B  X.  { 1o }
)  e.  _V
14 breq1 3938 . . . . 5  |-  ( x  =  ( A  X.  { (/) } )  -> 
( x  ~~  A  <->  ( A  X.  { (/) } )  ~~  A ) )
15 breq1 3938 . . . . 5  |-  ( y  =  ( B  X.  { 1o } )  -> 
( y  ~~  B  <->  ( B  X.  { 1o } )  ~~  B
) )
1614, 15bi2anan9 596 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  ~~  A  /\  y  ~~  B
)  <->  ( ( A  X.  { (/) } ) 
~~  A  /\  ( B  X.  { 1o }
)  ~~  B )
) )
17 ineq12 3275 . . . . 5  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( x  i^i  y
)  =  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) ) )
1817eqeq1d 2149 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  i^i  y )  =  (/)  <->  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) ) )
1916, 18anbi12d 465 . . 3  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )  <->  ( (
( A  X.  { (/)
} )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)  /\  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o } ) )  =  (/) ) ) )
2011, 13, 19spc2ev 2784 . 2  |-  ( ( ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B )  /\  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) )  ->  E. x E. y
( ( x  ~~  A  /\  y  ~~  B
)  /\  ( x  i^i  y )  =  (/) ) )
218, 9, 20mp2an 423 1  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   _Vcvv 2689    i^i cin 3073   (/)c0 3366   {csn 3530   class class class wbr 3935   Oncon0 4291    X. cxp 4543   1oc1o 6312    ~~ cen 6638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-iord 4294  df-on 4296  df-suc 4299  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-1o 6319  df-en 6641
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator