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Theorem endisj 6834
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 4047 . . . 4  |-  (/)  e.  _V
31, 2xpsnen 6831 . . 3  |-  ( A  X.  { (/) } ) 
~~  A
4 endisj.2 . . . 4  |-  B  e. 
_V
5 1on 6372 . . . . 5  |-  1o  e.  On
65elexi 2736 . . . 4  |-  1o  e.  _V
74, 6xpsnen 6831 . . 3  |-  ( B  X.  { 1o }
)  ~~  B
83, 7pm3.2i 443 . 2  |-  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)
9 xp01disj 6381 . 2  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
10 p0ex 4091 . . . 4  |-  { (/) }  e.  _V
111, 10xpex 4708 . . 3  |-  ( A  X.  { (/) } )  e.  _V
12 snex 4110 . . . 4  |-  { 1o }  e.  _V
134, 12xpex 4708 . . 3  |-  ( B  X.  { 1o }
)  e.  _V
14 breq1 3923 . . . . 5  |-  ( x  =  ( A  X.  { (/) } )  -> 
( x  ~~  A  <->  ( A  X.  { (/) } )  ~~  A ) )
15 breq1 3923 . . . . 5  |-  ( y  =  ( B  X.  { 1o } )  -> 
( y  ~~  B  <->  ( B  X.  { 1o } )  ~~  B
) )
1614, 15bi2anan9 848 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  ~~  A  /\  y  ~~  B
)  <->  ( ( A  X.  { (/) } ) 
~~  A  /\  ( B  X.  { 1o }
)  ~~  B )
) )
17 ineq12 3273 . . . . 5  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( x  i^i  y
)  =  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) ) )
1817eqeq1d 2261 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  i^i  y )  =  (/)  <->  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) ) )
1916, 18anbi12d 694 . . 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, 19cla42ev 2813 . 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 656 1  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285    X. cxp 4578   1oc1o 6358    ~~ cen 6746
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-1o 6365  df-en 6750
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