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

Theorem elssabg 4013
Description: Membership in a class abstraction involving a subset. Unlike elabg 2783,  A does not have to be a set. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elssabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elssabg  |-  ( B  e.  V  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elssabg
StepHypRef Expression
1 ssexg 4007 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
21expcom 115 . . 3  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  _V ) )
32adantrd 275 . 2  |-  ( B  e.  V  ->  (
( A  C_  B  /\  ps )  ->  A  e.  _V ) )
4 sseq1 3070 . . . 4  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
5 elssabg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5anbi12d 460 . . 3  |-  ( x  =  A  ->  (
( x  C_  B  /\  ph )  <->  ( A  C_  B  /\  ps )
) )
76elab3g 2788 . 2  |-  ( ( ( A  C_  B  /\  ps )  ->  A  e.  _V )  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
83, 7syl 14 1  |-  ( B  e.  V  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   {cab 2086   _Vcvv 2641    C_ wss 3021
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-v 2643  df-in 3027  df-ss 3034
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator