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Theorem unen 7610
Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
unen  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D )
)

Proof of Theorem unen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 7537 . . 3  |-  ( A 
~~  B  <->  E. x  x : A -1-1-onto-> B )
2 bren 7537 . . 3  |-  ( C 
~~  D  <->  E. y 
y : C -1-1-onto-> D )
3 eeanv 1957 . . . 4  |-  ( E. x E. y ( x : A -1-1-onto-> B  /\  y : C -1-1-onto-> D )  <->  ( E. x  x : A -1-1-onto-> B  /\  E. y  y : C -1-1-onto-> D
) )
4 vex 3121 . . . . . . . 8  |-  x  e. 
_V
5 vex 3121 . . . . . . . 8  |-  y  e. 
_V
64, 5unex 6593 . . . . . . 7  |-  ( x  u.  y )  e. 
_V
7 f1oun 5841 . . . . . . 7  |-  ( ( ( x : A -1-1-onto-> B  /\  y : C -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( x  u.  y ) : ( A  u.  C ) -1-1-onto-> ( B  u.  D ) )
8 f1oen3g 7543 . . . . . . 7  |-  ( ( ( x  u.  y
)  e.  _V  /\  ( x  u.  y
) : ( A  u.  C ) -1-1-onto-> ( B  u.  D ) )  ->  ( A  u.  C )  ~~  ( B  u.  D )
)
96, 7, 8sylancr 663 . . . . . 6  |-  ( ( ( x : A -1-1-onto-> B  /\  y : C -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D )
)
109ex 434 . . . . 5  |-  ( ( x : A -1-1-onto-> B  /\  y : C -1-1-onto-> D )  ->  (
( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~~  ( B  u.  D
) ) )
1110exlimivv 1699 . . . 4  |-  ( E. x E. y ( x : A -1-1-onto-> B  /\  y : C -1-1-onto-> D )  ->  (
( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~~  ( B  u.  D
) ) )
123, 11sylbir 213 . . 3  |-  ( ( E. x  x : A -1-1-onto-> B  /\  E. y 
y : C -1-1-onto-> D )  ->  ( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~~  ( B  u.  D
) ) )
131, 2, 12syl2anb 479 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( A  u.  C
)  ~~  ( B  u.  D ) ) )
1413imp 429 1  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118    u. cun 3479    i^i cin 3480   (/)c0 3790   class class class wbr 4453   -1-1-onto->wf1o 5593    ~~ cen 7525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-en 7529
This theorem is referenced by:  difsnen  7611  undom  7617  limensuci  7705  infensuc  7707  phplem2  7709  pssnn  7750  dif1enOLD  7764  dif1en  7765  unfi  7799  infdifsn  8085  pm54.43  8393  dif1card  8400  cdaun  8564  cdaen  8565  ssfin4  8702  fin23lem26  8717  unsnen  8940  fzennn  12058  mreexexlem4d  14919
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