ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssint Unicode version

Theorem ssint 3660
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  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
Distinct variable groups:    x, A    x, B

Proof of Theorem ssint
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfss3 2990 . 2  |-  ( A 
C_  |^| B  <->  A. y  e.  A  y  e.  |^| B )
2 vex 2605 . . . 4  |-  y  e. 
_V
32elint2 3651 . . 3  |-  ( y  e.  |^| B  <->  A. x  e.  B  y  e.  x )
43ralbii 2373 . 2  |-  ( A. y  e.  A  y  e.  |^| B  <->  A. y  e.  A  A. x  e.  B  y  e.  x )
5 ralcom 2518 . . 3  |-  ( A. y  e.  A  A. x  e.  B  y  e.  x  <->  A. x  e.  B  A. y  e.  A  y  e.  x )
6 dfss3 2990 . . . 4  |-  ( A 
C_  x  <->  A. y  e.  A  y  e.  x )
76ralbii 2373 . . 3  |-  ( A. x  e.  B  A  C_  x  <->  A. x  e.  B  A. y  e.  A  y  e.  x )
85, 7bitr4i 185 . 2  |-  ( A. y  e.  A  A. x  e.  B  y  e.  x  <->  A. x  e.  B  A  C_  x )
91, 4, 83bitri 204 1  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   A.wral 2349    C_ wss 2974   |^|cint 3644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-int 3645
This theorem is referenced by:  ssintab  3661  ssintub  3662  iinpw  3771  trint  3898  fintm  5106  bj-ssom  10889
  Copyright terms: Public domain W3C validator