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Theorem unen 6939
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 6867 . . 3  |-  ( A 
~~  B  <->  E. x  x : A -1-1-onto-> B )
2 bren 6867 . . 3  |-  ( C 
~~  D  <->  E. y 
y : C -1-1-onto-> D )
3 eeanv 1856 . . . 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 2792 . . . . . . . 8  |-  x  e. 
_V
5 vex 2792 . . . . . . . 8  |-  y  e. 
_V
64, 5unex 4517 . . . . . . 7  |-  ( x  u.  y )  e. 
_V
7 f1oun 5458 . . . . . . 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 6873 . . . . . . 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 644 . . . . . 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 423 . . . . 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 1668 . . . 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 204 . . 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 465 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( A  u.  C
)  ~~  ( B  u.  D ) ) )
1413imp 418 1  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   _Vcvv 2789    u. cun 3151    i^i cin 3152   (/)c0 3456   class class class wbr 4024   -1-1-onto->wf1o 5220    ~~ cen 6856
This theorem is referenced by:  difsnen  6940  undom  6946  limensuci  7033  infensuc  7035  phplem2  7037  pssnn  7077  dif1enOLD  7086  dif1en  7087  unfi  7120  infdifsn  7353  pm54.43  7629  dif1card  7634  cdaun  7794  cdaen  7795  ssfin4  7932  fin23lem26  7947  unsnen  8171  fzennn  11026  mreexexlem4d  13545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6860
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