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Theorem riinm 3720
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 3123 . 2  i^i  |^|_  X  S  |^|_  X  S  i^i
2 r19.2m 3303 . . . . 5  X  X  S  C_  X  S  C_
32ancoms 255 . . . 4  X  S  C_  X  X  S  C_
4 iinss 3699 . . . 4  X  S  C_  |^|_  X  S  C_
53, 4syl 14 . . 3  X  S  C_  X  |^|_  X  S  C_
6 df-ss 2925 . . 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 2081 1  X  S  C_  X  i^i  |^|_  X  S  |^|_  X  S
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301    i^i cin 2910    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-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651
This theorem is referenced by:  riinerm  6115
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