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

Theorem sseli 2941
 Description: Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
sseli.1 𝐴𝐵
Assertion
Ref Expression
sseli (𝐶𝐴𝐶𝐵)

Proof of Theorem sseli
StepHypRef Expression
1 sseli.1 . 2 𝐴𝐵
2 ssel 2939 . 2 (𝐴𝐵 → (𝐶𝐴𝐶𝐵))
31, 2ax-mp 7 1 (𝐶𝐴𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931 This theorem is referenced by:  sselii  2942  sseldi  2943  elun1  3110  elun2  3111  finds  4323  finds2  4324  issref  4707  2elresin  5010  fvun1  5239  fvmptssdm  5255  fvimacnvi  5281  elpreima  5286  ofrfval  5720  fnofval  5721  off  5724  offres  5762  eqopi  5798  op1steq  5805  dfoprab4  5818  reldmtpos  5868  smores3  5908  smores2  5909  pinn  6407  indpi  6440  enq0enq  6529  preqlu  6570  elinp  6572  prop  6573  elnp1st2nd  6574  prarloclem5  6598  cauappcvgprlemladd  6756  peano5nnnn  6966  nnindnn  6967  recn  7014  rexr  7071  peano5nni  7917  nnre  7921  nncn  7922  nnind  7930  nnnn0  8188  nn0re  8190  nn0cn  8191  nnz  8264  nn0z  8265  nnq  8568  qcn  8569  rpre  8589  iccshftri  8863  iccshftli  8865  iccdili  8867  icccntri  8869  fzval2  8877  fzelp1  8936  4fvwrd4  8997  elfzo1  9046  expcllem  9266  expcl2lemap  9267  m1expcl2  9277  cau3lem  9710  climconst2  9812
 Copyright terms: Public domain W3C validator