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Theorem riinm 3723
Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
riinm  X  S  C_  X  i^i  |^|_  X  S  |^|_  X  S
Distinct variable groups:   ,   , X
Allowed substitution hint:    S()

Proof of Theorem riinm
StepHypRef Expression
1 incom 3126 . 2  i^i  |^|_  X  S  |^|_  X  S  i^i
2 r19.2m 3306 . . . . 5  X  X  S  C_  X  S  C_
32ancoms 255 . . . 4  X  S  C_  X  X  S  C_
4 iinss 3702 . . . 4  X  S  C_  |^|_  X  S  C_
53, 4syl 14 . . 3  X  S  C_  X  |^|_  X  S  C_
6 df-ss 2928 . . 3  |^|_  X  S  C_  |^|_  X  S  i^i 
|^|_  X  S
75, 6sylib 127 . 2  X  S  C_  X  |^|_  X  S  i^i  |^|_  X  S
81, 7syl5eq 2084 1  X  S  C_  X  i^i  |^|_  X  S  |^|_  X  S
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1243  wex 1381   wcel 1393  wral 2303  wrex 2304    i^i cin 2913    C_ wss 2914   |^|_ciin 3652
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 2308  df-rex 2309  df-v 2556  df-in 2921  df-ss 2928  df-iin 3654
This theorem is referenced by:  riinerm  6119
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