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Theorem endisj 6298
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 3884 . . . 4  |-  (/)  e.  _V
31, 2xpsnen 6295 . . 3  |-  ( A  X.  { (/) } ) 
~~  A
4 endisj.2 . . . 4  |-  B  e. 
_V
5 1on 6008 . . . . 5  |-  1o  e.  On
65elexi 2567 . . . 4  |-  1o  e.  _V
74, 6xpsnen 6295 . . 3  |-  ( B  X.  { 1o }
)  ~~  B
83, 7pm3.2i 257 . 2  |-  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)
9 xp01disj 6017 . 2  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
10 p0ex 3939 . . . 4  |-  { (/) }  e.  _V
111, 10xpex 4453 . . 3  |-  ( A  X.  { (/) } )  e.  _V
126snex 3937 . . . 4  |-  { 1o }  e.  _V
134, 12xpex 4453 . . 3  |-  ( B  X.  { 1o }
)  e.  _V
14 breq1 3767 . . . . 5  |-  ( x  =  ( A  X.  { (/) } )  -> 
( x  ~~  A  <->  ( A  X.  { (/) } )  ~~  A ) )
15 breq1 3767 . . . . 5  |-  ( y  =  ( B  X.  { 1o } )  -> 
( y  ~~  B  <->  ( B  X.  { 1o } )  ~~  B
) )
1614, 15bi2anan9 538 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  ~~  A  /\  y  ~~  B
)  <->  ( ( A  X.  { (/) } ) 
~~  A  /\  ( B  X.  { 1o }
)  ~~  B )
) )
17 ineq12 3133 . . . . 5  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( x  i^i  y
)  =  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) ) )
1817eqeq1d 2048 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  i^i  y )  =  (/)  <->  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) ) )
1916, 18anbi12d 442 . . 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 2648 . 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 402 1  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557    i^i cin 2916   (/)c0 3224   {csn 3375   class class class wbr 3764   Oncon0 4100    X. cxp 4343   1oc1o 5994    ~~ cen 6219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-1o 6001  df-en 6222
This theorem is referenced by: (None)
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