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Theorem iununi 4167
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 2600 . . . . . . 7  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2 iunconst 4093 . . . . . . 7  |-  ( B  =/=  (/)  ->  U_ x  e.  B  A  =  A )
31, 2sylbir 205 . . . . . 6  |-  ( -.  B  =  (/)  ->  U_ x  e.  B  A  =  A )
4 iun0 4139 . . . . . . 7  |-  U_ x  e.  B  (/)  =  (/)
5 id 20 . . . . . . . 8  |-  ( A  =  (/)  ->  A  =  (/) )
65iuneq2d 4110 . . . . . . 7  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  U_ x  e.  B  (/) )
74, 6, 53eqtr4a 2493 . . . . . 6  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  A )
83, 7ja 155 . . . . 5  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  U_ x  e.  B  A  =  A )
98eqcomd 2440 . . . 4  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  A  =  U_ x  e.  B  A )
109uneq1d 3492 . . 3  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U_ x  e.  B  x )  =  ( U_ x  e.  B  A  u.  U_ x  e.  B  x
) )
11 uniiun 4136 . . . 4  |-  U. B  =  U_ x  e.  B  x
1211uneq2i 3490 . . 3  |-  ( A  u.  U. B )  =  ( A  u.  U_ x  e.  B  x )
13 iunun 4163 . . 3  |-  U_ x  e.  B  ( A  u.  x )  =  (
U_ x  e.  B  A  u.  U_ x  e.  B  x )
1410, 12, 133eqtr4g 2492 . 2  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
15 unieq 4016 . . . . . . 7  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
16 uni0 4034 . . . . . . 7  |-  U. (/)  =  (/)
1715, 16syl6eq 2483 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  (/) )
1817uneq2d 3493 . . . . 5  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
19 un0 3644 . . . . 5  |-  ( A  u.  (/) )  =  A
2018, 19syl6eq 2483 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
21 iuneq1 4098 . . . . 5  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
22 0iun 4140 . . . . 5  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
2321, 22syl6eq 2483 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
2420, 23eqeq12d 2449 . . 3  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
2524biimpcd 216 . 2  |-  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  ->  ( B  =  (/)  ->  A  =  (/) ) )
2614, 25impbii 181 1  |-  ( ( B  =  (/)  ->  A  =  (/) )  <->  ( A  u.  U. B )  = 
U_ x  e.  B  ( A  u.  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1652    =/= wne 2598    u. cun 3310   (/)c0 3620   U.cuni 4007   U_ciun 4085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-uni 4008  df-iun 4087
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