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Theorem iununir 3729
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iununir  u.  U.  U_  u.  (/)  (/)
Distinct variable groups:   ,   ,

Proof of Theorem iununir
StepHypRef Expression
1 unieq 3580 . . . . . 6  (/)  U.  U. (/)
2 uni0 3598 . . . . . 6  U. (/)  (/)
31, 2syl6eq 2085 . . . . 5  (/)  U.  (/)
43uneq2d 3091 . . . 4  (/)  u.  U.  u.  (/)
5 un0 3245 . . . 4  u.  (/)
64, 5syl6eq 2085 . . 3  (/)  u.  U.
7 iuneq1 3661 . . . 4  (/)  U_  u.  U_  (/)  u.
8 0iun 3705 . . . 4  U_  (/)  u.  (/)
97, 8syl6eq 2085 . . 3  (/)  U_  u.  (/)
106, 9eqeq12d 2051 . 2  (/)  u.  U.  U_  u.  (/)
1110biimpcd 148 1  u.  U.  U_  u.  (/)  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242    u. cun 2909   (/)c0 3218   U.cuni 3571   U_ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572  df-iun 3650
This theorem is referenced by: (None)
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