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Theorem iunun 3725
Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunun  U_  u.  C  U_  u.  U_  C

Proof of Theorem iunun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.43 2462 . . . 4  C  C
2 elun 3078 . . . . 5  u.  C  C
32rexbii 2325 . . . 4  u.  C  C
4 eliun 3652 . . . . 5  U_
5 eliun 3652 . . . . 5  U_  C  C
64, 5orbi12i 680 . . . 4  U_  U_  C  C
71, 3, 63bitr4i 201 . . 3  u.  C  U_  U_  C
8 eliun 3652 . . 3  U_  u.  C  u.  C
9 elun 3078 . . 3  U_  u.  U_  C  U_  U_  C
107, 8, 93bitr4i 201 . 2  U_  u.  C  U_  u.  U_  C
1110eqriv 2034 1  U_  u.  C  U_  u.  U_  C
Colors of variables: wff set class
Syntax hints:   wo 628   wceq 1242   wcel 1390  wrex 2301    u. cun 2909   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-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-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-un 2916  df-iun 3650
This theorem is referenced by: (None)
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