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Theorem iununir 3767
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iununir  |-  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  ->  ( B  =  (/)  ->  A  =  (/) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem iununir
StepHypRef Expression
1 unieq 3618 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
2 uni0 3636 . . . . . 6  |-  U. (/)  =  (/)
31, 2syl6eq 2130 . . . . 5  |-  ( B  =  (/)  ->  U. B  =  (/) )
43uneq2d 3127 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
5 un0 3285 . . . 4  |-  ( A  u.  (/) )  =  A
64, 5syl6eq 2130 . . 3  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
7 iuneq1 3699 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
8 0iun 3743 . . . 4  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
97, 8syl6eq 2130 . . 3  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
106, 9eqeq12d 2096 . 2  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
1110biimpcd 157 1  |-  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  ->  ( B  =  (/)  ->  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    u. cun 2972   (/)c0 3258   U.cuni 3609   U_ciun 3686
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  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-sn 3412  df-uni 3610  df-iun 3688
This theorem is referenced by: (None)
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