MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssab2 Structured version   Unicode version

Theorem ssab2 3584
Description: Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
Assertion
Ref Expression
ssab2  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssab2
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
21abssi 3575 1  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767   {cab 2452    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3483  df-ss 3490
This theorem is referenced by:  ssrab2  3585  zfausab  4596  exss  4710  dmopabss  5213  fabexg  6740  isf32lem9  8740  psubspset  34549  psubclsetN  34741
  Copyright terms: Public domain W3C validator