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Theorem unen 6703
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 6634 . . 3  |-  ( A 
~~  B  <->  E. x  x : A -1-1-onto-> B )
2 bren 6634 . . 3  |-  ( C 
~~  D  <->  E. y 
y : C -1-1-onto-> D )
3 eeanv 1902 . . . 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 2684 . . . . . . . 8  |-  x  e. 
_V
5 vex 2684 . . . . . . . 8  |-  y  e. 
_V
64, 5unex 4357 . . . . . . 7  |-  ( x  u.  y )  e. 
_V
7 f1oun 5380 . . . . . . 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 6641 . . . . . . 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 410 . . . . . 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 114 . . . . 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 1868 . . . 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 134 . . 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 289 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( A  u.  C
)  ~~  ( B  u.  D ) ) )
1413imp 123 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 103    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2681    u. cun 3064    i^i cin 3065   (/)c0 3358   class class class wbr 3924   -1-1-onto->wf1o 5117    ~~ cen 6625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-en 6628
This theorem is referenced by:  enpr2d  6704  phplem2  6740  fiunsnnn  6768  unsnfi  6800  endjusym  6974  pm54.43  7039  endjudisj  7059  djuen  7060  frecfzennn  10192  unennn  11899
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