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Theorem iunss 3689
Description: Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss  U_  C_  C  C_  C
Distinct variable group:   , C
Allowed substitution hints:   ()   ()

Proof of Theorem iunss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-iun 3650 . . 3  U_  {  |  }
21sseq1i 2963 . 2  U_  C_  C  {  |  }  C_  C
3 abss 3003 . 2  {  |  }  C_  C  C
4 dfss2 2928 . . . 4 
C_  C  C
54ralbii 2324 . . 3  C_  C  C
6 ralcom4 2570 . . 3  C  C
7 r19.23v 2419 . . . 4  C  C
87albii 1356 . . 3  C  C
95, 6, 83bitrri 196 . 2  C  C_  C
102, 3, 93bitri 195 1  U_  C_  C  C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98  wal 1240   wcel 1390   {cab 2023  wral 2300  wrex 2301    C_ wss 2911   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-bnd 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-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by:  iunss2  3693  djussxp  4424  fun11iun  5090
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