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Theorem ssrab2 3052
Description: Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
Assertion
Ref Expression
ssrab2 {𝑥𝐴𝜑} ⊆ 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrab2
StepHypRef Expression
1 df-rab 2332 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 ssab2 3051 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
31, 2eqsstri 3002 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wa 101  wcel 1409  {cab 2042  {crab 2327  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-in 2951  df-ss 2958
This theorem is referenced by:  ssrabeq  3053  rabexg  3927  pwnss  3939  onintrab2im  4271  ordtriexmidlem  4272  ontr2exmid  4277  ordtri2or2exmidlem  4278  onsucsssucexmid  4279  onsucelsucexmidlem  4281  tfis  4333  nnregexmid  4369  dmmptss  4844  ssimaex  5261  f1oresrab  5356  riotacl  5509  ssfiexmid  6366  genpelxp  6666  ltexprlempr  6763  cauappcvgprlemcl  6808  cauappcvgprlemladd  6813  caucvgprlemcl  6831  caucvgprprlemcl  6859  uzf  8571  rpre  8686  ixxf  8867  fzf  8979  serige0  9411  expcl2lemap  9426  expclzaplem  9438  expge0  9450  expge1  9451  dvdsflip  10155  bdrabexg  10385
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