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Theorem iununi 3927
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iununi  |-  ( ( B  =  (/)  ->  A  =  (/) )  <->  ( A  u.  U. B )  = 
U_ x  e.  B  ( A  u.  x
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem iununi
StepHypRef Expression
1 df-ne 2421 . . . . . . 7  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2 iunconst 3854 . . . . . . 7  |-  ( B  =/=  (/)  ->  U_ x  e.  B  A  =  A )
31, 2sylbir 206 . . . . . 6  |-  ( -.  B  =  (/)  ->  U_ x  e.  B  A  =  A )
4 iun0 3899 . . . . . . 7  |-  U_ x  e.  B  (/)  =  (/)
5 id 21 . . . . . . . 8  |-  ( A  =  (/)  ->  A  =  (/) )
65iuneq2d 3871 . . . . . . 7  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  U_ x  e.  B  (/) )
74, 6, 53eqtr4a 2314 . . . . . 6  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  A )
83, 7ja 155 . . . . 5  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  U_ x  e.  B  A  =  A )
98eqcomd 2261 . . . 4  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  A  =  U_ x  e.  B  A )
109uneq1d 3270 . . 3  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U_ x  e.  B  x )  =  ( U_ x  e.  B  A  u.  U_ x  e.  B  x
) )
11 uniiun 3896 . . . 4  |-  U. B  =  U_ x  e.  B  x
1211uneq2i 3268 . . 3  |-  ( A  u.  U. B )  =  ( A  u.  U_ x  e.  B  x )
13 iunun 3923 . . 3  |-  U_ x  e.  B  ( A  u.  x )  =  (
U_ x  e.  B  A  u.  U_ x  e.  B  x )
1410, 12, 133eqtr4g 2313 . 2  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
15 unieq 3777 . . . . . . 7  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
16 uni0 3795 . . . . . . 7  |-  U. (/)  =  (/)
1715, 16syl6eq 2304 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  (/) )
1817uneq2d 3271 . . . . 5  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
19 un0 3421 . . . . 5  |-  ( A  u.  (/) )  =  A
2018, 19syl6eq 2304 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
21 iuneq1 3859 . . . . 5  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
22 0iun 3900 . . . . 5  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
2321, 22syl6eq 2304 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
2420, 23eqeq12d 2270 . . 3  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
2524biimpcd 217 . 2  |-  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  ->  ( B  =  (/)  ->  A  =  (/) ) )
2614, 25impbii 182 1  |-  ( ( B  =  (/)  ->  A  =  (/) )  <->  ( A  u.  U. B )  = 
U_ x  e.  B  ( A  u.  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    = wceq 1619    =/= wne 2419    u. cun 3092   (/)c0 3397   U.cuni 3768   U_ciun 3846
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-sn 3587  df-uni 3769  df-iun 3848
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