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Theorem iinuniss 3728
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iinuniss  u.  |^|  C_  |^|_  u.
Distinct variable groups:   ,   ,

Proof of Theorem iinuniss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.32vr 2452 . . . 4
2 vex 2554 . . . . . 6 
_V
32elint2 3613 . . . . 5  |^|
43orbi2i 678 . . . 4  |^|
5 elun 3078 . . . . 5  u.
65ralbii 2324 . . . 4  u.
71, 4, 63imtr4i 190 . . 3  |^|  u.
87ss2abi 3006 . 2  {  | 
|^| }  C_ 
{  |  u.  }
9 df-un 2916 . 2  u.  |^|  {  |  |^| }
10 df-iin 3651 . 2  |^|_  u.  {  |  u.  }
118, 9, 103sstr4i 2978 1  u.  |^|  C_  |^|_  u.
Colors of variables: wff set class
Syntax hints:   wo 628   wcel 1390   {cab 2023  wral 2300    u. cun 2909    C_ wss 2911   |^|cint 3606   |^|_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-un 2916  df-in 2918  df-ss 2925  df-int 3607  df-iin 3651
This theorem is referenced by: (None)
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