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Theorem sess1 4822
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 458 . . . . . 6  |-  ( ( R  C_  S  /\  y  e.  A )  ->  R  C_  S )
21ssbrd 4467 . . . . 5  |-  ( ( R  C_  S  /\  y  e.  A )  ->  ( y R x  ->  y S x ) )
32ss2rabdv 3548 . . . 4  |-  ( R 
C_  S  ->  { y  e.  A  |  y R x }  C_  { y  e.  A  | 
y S x }
)
4 ssexg 4571 . . . . 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 435 . . . 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 17 . . 3  |-  ( R 
C_  S  ->  ( { y  e.  A  |  y S x }  e.  _V  ->  { y  e.  A  | 
y R x }  e.  _V ) )
76ralimdv 2842 . 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 4814 . 2  |-  ( S Se  A  <->  A. x  e.  A  { y  e.  A  |  y S x }  e.  _V )
9 df-se 4814 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
107, 8, 93imtr4g 273 1  |-  ( R 
C_  S  ->  ( S Se  A  ->  R Se  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    C_ wss 3442   class class class wbr 4426   Se wse 4811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-in 3449  df-ss 3456  df-br 4427  df-se 4814
This theorem is referenced by:  seeq1  4826
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