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

Theorem sseld 3066
Description: Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
Hypothesis
Ref Expression
sseld.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
sseld  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )

Proof of Theorem sseld
StepHypRef Expression
1 sseld.1 . 2  |-  ( ph  ->  A  C_  B )
2 ssel 3061 . 2  |-  ( A 
C_  B  ->  ( C  e.  A  ->  C  e.  B ) )
31, 2syl 14 1  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  sselda  3067  sseldd  3068  ssneld  3069  elelpwi  3492  ssbrd  3941  uniopel  4148  onintonm  4403  sucprcreg  4434  ordsuc  4448  0elnn  4502  dmrnssfld  4772  nfunv  5126  opelf  5264  fvimacnv  5503  ffvelrn  5521  resflem  5552  f1imass  5643  dftpos3  6127  nnmordi  6380  mapsn  6552  ixpf  6582  diffifi  6756  ordiso2  6888  difinfinf  6954  prarloclemarch2  7195  ltexprlemrl  7386  cauappcvgprlemladdrl  7433  caucvgprlemladdrl  7454  caucvgprlem1  7455  axpre-suploclemres  7677  uzind  9130  supinfneg  9358  infsupneg  9359  ixxssxr  9651  elfz0add  9868  fzoss1  9916  frecuzrdgrclt  10156  fsum3cvg  11114  isumrpcl  11231  lmtopcnp  12346  txuni2  12352  tx1cn  12365  tx2cn  12366  txlm  12375  imasnopn  12395  xmetunirn  12454  mopnval  12538  metrest  12602  dedekindicc  12707  ivthdec  12718  limcimolemlt  12729  bj-nnord  13083
  Copyright terms: Public domain W3C validator