MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iununi Structured version   Unicode version

Theorem iununi 4331
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 2579 . . . . . . 7  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2 iunconst 4252 . . . . . . 7  |-  ( B  =/=  (/)  ->  U_ x  e.  B  A  =  A )
31, 2sylbir 213 . . . . . 6  |-  ( -.  B  =  (/)  ->  U_ x  e.  B  A  =  A )
4 iun0 4299 . . . . . . 7  |-  U_ x  e.  B  (/)  =  (/)
5 id 22 . . . . . . . 8  |-  ( A  =  (/)  ->  A  =  (/) )
65iuneq2d 4270 . . . . . . 7  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  U_ x  e.  B  (/) )
74, 6, 53eqtr4a 2449 . . . . . 6  |-  ( A  =  (/)  ->  U_ x  e.  B  A  =  A )
83, 7ja 161 . . . . 5  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  U_ x  e.  B  A  =  A )
98eqcomd 2390 . . . 4  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  A  =  U_ x  e.  B  A )
109uneq1d 3571 . . 3  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U_ x  e.  B  x )  =  ( U_ x  e.  B  A  u.  U_ x  e.  B  x
) )
11 uniiun 4296 . . . 4  |-  U. B  =  U_ x  e.  B  x
1211uneq2i 3569 . . 3  |-  ( A  u.  U. B )  =  ( A  u.  U_ x  e.  B  x )
13 iunun 4327 . . 3  |-  U_ x  e.  B  ( A  u.  x )  =  (
U_ x  e.  B  A  u.  U_ x  e.  B  x )
1410, 12, 133eqtr4g 2448 . 2  |-  ( ( B  =  (/)  ->  A  =  (/) )  ->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
15 unieq 4171 . . . . . . 7  |-  ( B  =  (/)  ->  U. B  =  U. (/) )
16 uni0 4190 . . . . . . 7  |-  U. (/)  =  (/)
1715, 16syl6eq 2439 . . . . . 6  |-  ( B  =  (/)  ->  U. B  =  (/) )
1817uneq2d 3572 . . . . 5  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  ( A  u.  (/) ) )
19 un0 3737 . . . . 5  |-  ( A  u.  (/) )  =  A
2018, 19syl6eq 2439 . . . 4  |-  ( B  =  (/)  ->  ( A  u.  U. B )  =  A )
21 iuneq1 4257 . . . . 5  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  U_ x  e.  (/)  ( A  u.  x ) )
22 0iun 4300 . . . . 5  |-  U_ x  e.  (/)  ( A  u.  x )  =  (/)
2321, 22syl6eq 2439 . . . 4  |-  ( B  =  (/)  ->  U_ x  e.  B  ( A  u.  x )  =  (/) )
2420, 23eqeq12d 2404 . . 3  |-  ( B  =  (/)  ->  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  <->  A  =  (/) ) )
2524biimpcd 224 . 2  |-  ( ( A  u.  U. B
)  =  U_ x  e.  B  ( A  u.  x )  ->  ( B  =  (/)  ->  A  =  (/) ) )
2614, 25impbii 188 1  |-  ( ( B  =  (/)  ->  A  =  (/) )  <->  ( A  u.  U. B )  = 
U_ x  e.  B  ( A  u.  x
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1399    =/= wne 2577    u. cun 3387   (/)c0 3711   U.cuni 4163   U_ciun 4243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-sn 3945  df-uni 4164  df-iun 4245
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator