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

Theorem unen 7152
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 7080 . . 3  |-  ( A 
~~  B  <->  E. x  x : A -1-1-onto-> B )
2 bren 7080 . . 3  |-  ( C 
~~  D  <->  E. y 
y : C -1-1-onto-> D )
3 eeanv 1933 . . . 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 2923 . . . . . . . 8  |-  x  e. 
_V
5 vex 2923 . . . . . . . 8  |-  y  e. 
_V
64, 5unex 4670 . . . . . . 7  |-  ( x  u.  y )  e. 
_V
7 f1oun 5657 . . . . . . 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 7086 . . . . . . 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 645 . . . . . 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 424 . . . . 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 1642 . . . 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 205 . . 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 466 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( A  u.  C
)  ~~  ( B  u.  D ) ) )
1413imp 419 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2920    u. cun 3282    i^i cin 3283   (/)c0 3592   class class class wbr 4176   -1-1-onto->wf1o 5416    ~~ cen 7069
This theorem is referenced by:  difsnen  7153  undom  7159  limensuci  7246  infensuc  7248  phplem2  7250  pssnn  7290  dif1enOLD  7303  dif1en  7304  unfi  7337  infdifsn  7571  pm54.43  7847  dif1card  7852  cdaun  8012  cdaen  8013  ssfin4  8150  fin23lem26  8165  unsnen  8388  fzennn  11266  mreexexlem4d  13831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-en 7073
  Copyright terms: Public domain W3C validator