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Theorem iununi 3960
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 2423 . . . . . . 7  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2 iunconst 3887 . . . . . . 7  |-  ( B  =/=  (/)  ->  U_ x  e.  B  A  =  A )
31, 2sylbir 206 . . . . . 6  |-  ( -.  B  =  (/)  ->  U_ x  e.  B  A  =  A )
4 iun0 3932 . . . . . . 7  |-  U_ x  e.  B  (/)  =  (/)
5 id 21 . . . . . . . 8  |-  ( A  =  (/)  ->  A  =  (/) )
65iuneq2d 3904 . . . . . . 7  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  U_ x  e.  B  (/) )
74, 6, 53eqtr4a 2316 . . . . . 6  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  A )
83, 7ja 155 . . . . 5  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  U_ x  e.  B  A  =  A )
98eqcomd 2263 . . . 4  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  A  =  U_ x  e.  B  A )
109uneq1d 3303 . . 3  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U_ x  e.  B  x )  =  ( U_ x  e.  B  A  u.  U_ x  e.  B  x
) )
11 uniiun 3929 . . . 4  |-  U. B  =  U_ x  e.  B  x
1211uneq2i 3301 . . 3  |-  ( A  u.  U. B )  =  ( A  u.  U_ x  e.  B  x )
13 iunun 3956 . . 3  |-  U_ x  e.  B  ( A  u.  x )  =  (
U_ x  e.  B  A  u.  U_ x  e.  B  x )
1410, 12, 133eqtr4g 2315 . 2  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
15 unieq 3810 . . . . . . 7  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
16 uni0 3828 . . . . . . 7  |-  U. (/)  =  (/)
1715, 16syl6eq 2306 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  (/) )
1817uneq2d 3304 . . . . 5  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
19 un0 3454 . . . . 5  |-  ( A  u.  (/) )  =  A
2018, 19syl6eq 2306 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
21 iuneq1 3892 . . . . 5  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
22 0iun 3933 . . . . 5  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
2321, 22syl6eq 2306 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
2420, 23eqeq12d 2272 . . 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 2421    u. cun 3125   (/)c0 3430   U.cuni 3801   U_ciun 3879
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 2239
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 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-sn 3620  df-uni 3802  df-iun 3881
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