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Theorem iununi 4135
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 2569 . . . . . . 7  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2 iunconst 4061 . . . . . . 7  |-  ( B  =/=  (/)  ->  U_ x  e.  B  A  =  A )
31, 2sylbir 205 . . . . . 6  |-  ( -.  B  =  (/)  ->  U_ x  e.  B  A  =  A )
4 iun0 4107 . . . . . . 7  |-  U_ x  e.  B  (/)  =  (/)
5 id 20 . . . . . . . 8  |-  ( A  =  (/)  ->  A  =  (/) )
65iuneq2d 4078 . . . . . . 7  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  U_ x  e.  B  (/) )
74, 6, 53eqtr4a 2462 . . . . . 6  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  A )
83, 7ja 155 . . . . 5  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  U_ x  e.  B  A  =  A )
98eqcomd 2409 . . . 4  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  A  =  U_ x  e.  B  A )
109uneq1d 3460 . . 3  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U_ x  e.  B  x )  =  ( U_ x  e.  B  A  u.  U_ x  e.  B  x
) )
11 uniiun 4104 . . . 4  |-  U. B  =  U_ x  e.  B  x
1211uneq2i 3458 . . 3  |-  ( A  u.  U. B )  =  ( A  u.  U_ x  e.  B  x )
13 iunun 4131 . . 3  |-  U_ x  e.  B  ( A  u.  x )  =  (
U_ x  e.  B  A  u.  U_ x  e.  B  x )
1410, 12, 133eqtr4g 2461 . 2  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
15 unieq 3984 . . . . . . 7  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
16 uni0 4002 . . . . . . 7  |-  U. (/)  =  (/)
1715, 16syl6eq 2452 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  (/) )
1817uneq2d 3461 . . . . 5  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
19 un0 3612 . . . . 5  |-  ( A  u.  (/) )  =  A
2018, 19syl6eq 2452 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
21 iuneq1 4066 . . . . 5  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
22 0iun 4108 . . . . 5  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
2321, 22syl6eq 2452 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
2420, 23eqeq12d 2418 . . 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 1649    =/= wne 2567    u. cun 3278   (/)c0 3588   U.cuni 3975   U_ciun 4053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780  df-uni 3976  df-iun 4055
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