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

Theorem riinm 3785
Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
riinm  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Distinct variable groups:    x, A    x, X
Allowed substitution hint:    S( x)

Proof of Theorem riinm
StepHypRef Expression
1 incom 3181 . 2  |-  ( A  i^i  |^|_ x  e.  X  S )  =  (
|^|_ x  e.  X  S  i^i  A )
2 r19.2m 3356 . . . . 5  |-  ( ( E. x  x  e.  X  /\  A. x  e.  X  S  C_  A
)  ->  E. x  e.  X  S  C_  A
)
32ancoms 264 . . . 4  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  E. x  e.  X  S  C_  A
)
4 iinss 3764 . . . 4  |-  ( E. x  e.  X  S  C_  A  ->  |^|_ x  e.  X  S  C_  A
)
53, 4syl 14 . . 3  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  |^|_ x  e.  X  S  C_  A
)
6 df-ss 3001 . . 3  |-  ( |^|_ x  e.  X  S  C_  A 
<->  ( |^|_ x  e.  X  S  i^i  A )  = 
|^|_ x  e.  X  S )
75, 6sylib 120 . 2  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( |^|_ x  e.  X  S  i^i  A )  =  |^|_ x  e.  X  S )
81, 7syl5eq 2129 1  |-  ( ( A. x  e.  X  S  C_  A  /\  E. x  x  e.  X
)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  |^|_ x  e.  X  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287   E.wex 1424    e. wcel 1436   A.wral 2355   E.wrex 2356    i^i cin 2987    C_ wss 2988   |^|_ciin 3714
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-iin 3716
This theorem is referenced by:  riinerm  6317
  Copyright terms: Public domain W3C validator