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Theorem iununir 3967
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 3816 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
2 uni0 3834 . . . . . 6  |-  U. (/)  =  (/)
31, 2eqtrdi 2226 . . . . 5  |-  ( B  =  (/)  ->  U. B  =  (/) )
43uneq2d 3289 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
5 un0 3456 . . . 4  |-  ( A  u.  (/) )  =  A
64, 5eqtrdi 2226 . . 3  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
7 iuneq1 3897 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
8 0iun 3941 . . . 4  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
97, 8eqtrdi 2226 . . 3  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
106, 9eqeq12d 2192 . 2  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
1110biimpcd 159 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 1353    u. cun 3127   (/)c0 3422   U.cuni 3807   U_ciun 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-uni 3808  df-iun 3886
This theorem is referenced by: (None)
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