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

Theorem ssrin 3271
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssrin  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )

Proof of Theorem ssrin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3061 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 334 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  x  e.  C
)  ->  ( x  e.  B  /\  x  e.  C ) ) )
3 elin 3229 . . 3  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3229 . . 3  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
52, 3, 43imtr4g 204 . 2  |-  ( A 
C_  B  ->  (
x  e.  ( A  i^i  C )  ->  x  e.  ( B  i^i  C ) ) )
65ssrdv 3073 1  |-  ( A 
C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465    i^i cin 3040    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by:  sslin  3272  ssrind  3273  ss2in  3274  ssdisj  3389  ssdifin0  3414  ssres  4815  phplem2  6715  sbthlem7  6819  fiss  6833  tgss  12159  metrest  12602
  Copyright terms: Public domain W3C validator