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Theorem iununir 3866
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 3715 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
2 uni0 3733 . . . . . 6  |-  U. (/)  =  (/)
31, 2syl6eq 2166 . . . . 5  |-  ( B  =  (/)  ->  U. B  =  (/) )
43uneq2d 3200 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
5 un0 3366 . . . 4  |-  ( A  u.  (/) )  =  A
64, 5syl6eq 2166 . . 3  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
7 iuneq1 3796 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
8 0iun 3840 . . . 4  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
97, 8syl6eq 2166 . . 3  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
106, 9eqeq12d 2132 . 2  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
1110biimpcd 158 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 1316    u. cun 3039   (/)c0 3333   U.cuni 3706   U_ciun 3783
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-uni 3707  df-iun 3785
This theorem is referenced by: (None)
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