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Theorem iunxun 3735
 Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun

Proof of Theorem iunxun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rexun 3123 . . . 4
2 eliun 3661 . . . . 5
3 eliun 3661 . . . . 5
42, 3orbi12i 681 . . . 4
51, 4bitr4i 176 . . 3
6 eliun 3661 . . 3
7 elun 3084 . . 3
85, 6, 73bitr4i 201 . 2
98eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wo 629   wceq 1243   wcel 1393  wrex 2307   cun 2915  ciun 3657 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-iun 3659 This theorem is referenced by:  iunsuc  4157  rdgisuc1  5971  oasuc  6044  omsuc  6051
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