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Theorem ssint 3622
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint 
C_  |^|  C_
Distinct variable groups:   ,   ,

Proof of Theorem ssint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfss3 2929 . 2 
C_  |^|  |^|
2 vex 2554 . . . 4 
_V
32elint2 3613 . . 3  |^|
43ralbii 2324 . 2  |^|
5 ralcom 2467 . . 3
6 dfss3 2929 . . . 4 
C_
76ralbii 2324 . . 3  C_
85, 7bitr4i 176 . 2  C_
91, 4, 83bitri 195 1 
C_  |^|  C_
Colors of variables: wff set class
Syntax hints:   wb 98   wcel 1390  wral 2300    C_ wss 2911   |^|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-bnd 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-int 3607
This theorem is referenced by:  ssintab  3623  ssintub  3624  iinpw  3733  trint  3860  fintm  5018  bj-ssom  9324
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