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Theorem unen 6361
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 6294 . . 3  |-  ( A 
~~  B  <->  E. x  x : A -1-1-onto-> B )
2 bren 6294 . . 3  |-  ( C 
~~  D  <->  E. y 
y : C -1-1-onto-> D )
3 eeanv 1849 . . . 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 2605 . . . . . . . 8  |-  x  e. 
_V
5 vex 2605 . . . . . . . 8  |-  y  e. 
_V
64, 5unex 4202 . . . . . . 7  |-  ( x  u.  y )  e. 
_V
7 f1oun 5177 . . . . . . 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 6301 . . . . . . 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 405 . . . . . 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 113 . . . . 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 1818 . . . 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 133 . . 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 285 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( A  u.  C
)  ~~  ( B  u.  D ) ) )
1413imp 122 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 102    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602    u. cun 2972    i^i cin 2973   (/)c0 3258   class class class wbr 3793   -1-1-onto->wf1o 4931    ~~ cen 6285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-en 6288
This theorem is referenced by:  phplem2  6388  fiunsnnn  6415  unsnfi  6439  pm54.43  6518  frecfzennn  9508  unennn  10708
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