ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sess1 Unicode version

Theorem sess1 4197
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )

Proof of Theorem sess1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . . . . 6  |-  ( ( R  C_  S  /\  y  e.  A )  ->  R  C_  S )
21ssbrd 3916 . . . . 5  |-  ( ( R  C_  S  /\  y  e.  A )  ->  ( y R x  ->  y S x ) )
32ss2rabdv 3125 . . . 4  |-  ( R 
C_  S  ->  { y  e.  A  |  y R x }  C_  { y  e.  A  | 
y S x }
)
4 ssexg 4007 . . . . 5  |-  ( ( { y  e.  A  |  y R x }  C_  { y  e.  A  |  y S x }  /\  { y  e.  A  | 
y S x }  e.  _V )  ->  { y  e.  A  |  y R x }  e.  _V )
54ex 114 . . . 4  |-  ( { y  e.  A  | 
y R x }  C_ 
{ y  e.  A  |  y S x }  ->  ( {
y  e.  A  | 
y S x }  e.  _V  ->  { y  e.  A  |  y R x }  e.  _V ) )
63, 5syl 14 . . 3  |-  ( R 
C_  S  ->  ( { y  e.  A  |  y S x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
76ralimdv 2459 . 2  |-  ( R 
C_  S  ->  ( A. x  e.  A  { y  e.  A  |  y S x }  e.  _V  ->  A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V ) )
8 df-se 4193 . 2  |-  ( S Se  A  <->  A. x  e.  A  { y  e.  A  |  y S x }  e.  _V )
9 df-se 4193 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 204 1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1448   A.wral 2375   {crab 2379   _Vcvv 2641    C_ wss 3021   class class class wbr 3875   Se wse 4189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384  df-v 2643  df-in 3027  df-ss 3034  df-br 3876  df-se 4193
This theorem is referenced by:  seeq1  4199
  Copyright terms: Public domain W3C validator