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Theorem endisj 6817
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 4127 . . . 4  |-  (/)  e.  _V
31, 2xpsnen 6814 . . 3  |-  ( A  X.  { (/) } ) 
~~  A
4 endisj.2 . . . 4  |-  B  e. 
_V
5 1on 6417 . . . . 5  |-  1o  e.  On
65elexi 2749 . . . 4  |-  1o  e.  _V
74, 6xpsnen 6814 . . 3  |-  ( B  X.  { 1o }
)  ~~  B
83, 7pm3.2i 272 . 2  |-  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)
9 xp01disj 6427 . 2  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
10 p0ex 4185 . . . 4  |-  { (/) }  e.  _V
111, 10xpex 4737 . . 3  |-  ( A  X.  { (/) } )  e.  _V
126snex 4182 . . . 4  |-  { 1o }  e.  _V
134, 12xpex 4737 . . 3  |-  ( B  X.  { 1o }
)  e.  _V
14 breq1 4003 . . . . 5  |-  ( x  =  ( A  X.  { (/) } )  -> 
( x  ~~  A  <->  ( A  X.  { (/) } )  ~~  A ) )
15 breq1 4003 . . . . 5  |-  ( y  =  ( B  X.  { 1o } )  -> 
( y  ~~  B  <->  ( B  X.  { 1o } )  ~~  B
) )
1614, 15bi2anan9 606 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  ~~  A  /\  y  ~~  B
)  <->  ( ( A  X.  { (/) } ) 
~~  A  /\  ( B  X.  { 1o }
)  ~~  B )
) )
17 ineq12 3331 . . . . 5  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( x  i^i  y
)  =  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) ) )
1817eqeq1d 2186 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  i^i  y )  =  (/)  <->  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) ) )
1916, 18anbi12d 473 . . 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 2833 . 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 426 1  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   _Vcvv 2737    i^i cin 3128   (/)c0 3422   {csn 3591   class class class wbr 4000   Oncon0 4359    X. cxp 4620   1oc1o 6403    ~~ cen 6731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-1o 6410  df-en 6734
This theorem is referenced by: (None)
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