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Theorem ssrab2 3080
Description: Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
Assertion
Ref Expression
ssrab2  |-  { x  e.  A  |  ph }  C_  A
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssrab2
StepHypRef Expression
1 df-rab 2358 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 ssab2 3079 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
31, 2eqsstri 3030 1  |-  { x  e.  A  |  ph }  C_  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1434   {cab 2068   {crab 2353    C_ wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-in 2980  df-ss 2987
This theorem is referenced by:  ssrabeq  3081  rabexg  3929  pwnss  3941  onintrab2im  4270  ordtriexmidlem  4271  ontr2exmid  4276  ordtri2or2exmidlem  4277  onsucsssucexmid  4278  onsucelsucexmidlem  4280  tfis  4332  nnregexmid  4368  dmmptss  4847  ssimaex  5266  f1oresrab  5361  riotacl  5513  ssfiexmid  6411  domfiexmid  6413  genpelxp  6763  ltexprlempr  6860  cauappcvgprlemcl  6905  cauappcvgprlemladd  6910  caucvgprlemcl  6928  caucvgprprlemcl  6956  uzf  8703  supminfex  8766  rpre  8821  ixxf  8997  fzf  9109  serige0  9570  expcl2lemap  9585  expclzaplem  9597  expge0  9609  expge1  9610  dvdsflip  10396  infssuzex  10489  infssuzcldc  10491  gcddvds  10499  lcmn0cl  10594  bdrabexg  10855
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