MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  endisj Structured version   Unicode version

Theorem endisj 7597
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 4569 . . . 4  |-  (/)  e.  _V
31, 2xpsnen 7594 . . 3  |-  ( A  X.  { (/) } ) 
~~  A
4 endisj.2 . . . 4  |-  B  e. 
_V
5 1on 7129 . . . . 5  |-  1o  e.  On
65elexi 3116 . . . 4  |-  1o  e.  _V
74, 6xpsnen 7594 . . 3  |-  ( B  X.  { 1o }
)  ~~  B
83, 7pm3.2i 453 . 2  |-  ( ( A  X.  { (/) } )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)
9 xp01disj 7138 . 2  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
10 p0ex 4624 . . . 4  |-  { (/) }  e.  _V
111, 10xpex 6577 . . 3  |-  ( A  X.  { (/) } )  e.  _V
12 snex 4678 . . . 4  |-  { 1o }  e.  _V
134, 12xpex 6577 . . 3  |-  ( B  X.  { 1o }
)  e.  _V
14 breq1 4442 . . . . 5  |-  ( x  =  ( A  X.  { (/) } )  -> 
( x  ~~  A  <->  ( A  X.  { (/) } )  ~~  A ) )
15 breq1 4442 . . . . 5  |-  ( y  =  ( B  X.  { 1o } )  -> 
( y  ~~  B  <->  ( B  X.  { 1o } )  ~~  B
) )
1614, 15bi2anan9 871 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  ~~  A  /\  y  ~~  B
)  <->  ( ( A  X.  { (/) } ) 
~~  A  /\  ( B  X.  { 1o }
)  ~~  B )
) )
17 ineq12 3681 . . . . 5  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( x  i^i  y
)  =  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) ) )
1817eqeq1d 2456 . . . 4  |-  ( ( x  =  ( A  X.  { (/) } )  /\  y  =  ( B  X.  { 1o } ) )  -> 
( ( x  i^i  y )  =  (/)  <->  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) ) )
1916, 18anbi12d 708 . . 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 3199 . 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 670 1  |-  E. x E. y ( ( x 
~~  A  /\  y  ~~  B )  /\  (
x  i^i  y )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    i^i cin 3460   (/)c0 3783   {csn 4016   class class class wbr 4439   Oncon0 4867    X. cxp 4986   1oc1o 7115    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-1o 7122  df-en 7510
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator