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

Theorem sselda 2942
Description: Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
Hypothesis
Ref Expression
sseld.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
sselda  |-  ( (
ph  /\  C  e.  A )  ->  C  e.  B )

Proof of Theorem sselda
StepHypRef Expression
1 sseld.1 . . 3  |-  ( ph  ->  A  C_  B )
21sseld 2941 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32imp 115 1  |-  ( (
ph  /\  C  e.  A )  ->  C  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393    C_ wss 2914
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 2921  df-ss 2928
This theorem is referenced by:  elrel  4420  ffvresb  5306  1stdm  5786  tfrlem1  5901  tfrlemiubacc  5922  erinxp  6158  fundmen  6264  elprnql  6551  elprnqu  6552  un0addcl  8178  un0mulcl  8179  icoshftf1o  8817  elfzom1elfzo  9017  zpnn0elfzo  9021  iseqfveq  9099  monoord2  9105
  Copyright terms: Public domain W3C validator