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Theorem iinss1 3660
Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1 
C_  |^|_  C  C_  |^|_  C
Distinct variable groups:   ,   ,
Allowed substitution hint:    C()

Proof of Theorem iinss1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssralv 2998 . . 3 
C_  C  C
2 vex 2554 . . . 4 
_V
3 eliin 3653 . . . 4  _V  |^|_  C  C
42, 3ax-mp 7 . . 3  |^|_  C  C
5 eliin 3653 . . . 4  _V  |^|_  C  C
62, 5ax-mp 7 . . 3  |^|_  C  C
71, 4, 63imtr4g 194 . 2 
C_  |^|_  C  |^|_  C
87ssrdv 2945 1 
C_  |^|_  C  C_  |^|_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wcel 1390  wral 2300   _Vcvv 2551    C_ wss 2911   |^|_ciin 3649
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-v 2553  df-in 2918  df-ss 2925  df-iin 3651
This theorem is referenced by: (None)
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