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Theorem rintm 3735
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
rintm  X  C_  ~P  X  i^i  |^| X 
|^| X
Distinct variable group:   , X
Allowed substitution hint:   ()

Proof of Theorem rintm
StepHypRef Expression
1 incom 3123 . 2  i^i  |^| X  |^| X  i^i
2 intssuni2m 3630 . . . 4  X  C_  ~P  X  |^| X  C_  U. ~P
3 ssid 2958 . . . . 5  ~P  C_ 
~P
4 sspwuni 3730 . . . . 5  ~P  C_  ~P  U. ~P  C_
53, 4mpbi 133 . . . 4  U. ~P  C_
62, 5syl6ss 2951 . . 3  X  C_  ~P  X  |^| X  C_
7 df-ss 2925 . . 3  |^| X  C_  |^| X  i^i 
|^| X
86, 7sylib 127 . 2  X  C_  ~P  X  |^| X  i^i  |^| X
91, 8syl5eq 2081 1  X  C_  ~P  X  i^i  |^| X 
|^| X
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390    i^i cin 2910    C_ wss 2911   ~Pcpw 3351   U.cuni 3571   |^|cint 3606
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-pw 3353  df-uni 3572  df-int 3607
This theorem is referenced by: (None)
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