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

Theorem sseldd 3013
Description: Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
Hypotheses
Ref Expression
sseld.1  |-  ( ph  ->  A  C_  B )
sseldd.2  |-  ( ph  ->  C  e.  A )
Assertion
Ref Expression
sseldd  |-  ( ph  ->  C  e.  B )

Proof of Theorem sseldd
StepHypRef Expression
1 sseldd.2 . 2  |-  ( ph  ->  C  e.  A )
2 sseld.1 . . 3  |-  ( ph  ->  A  C_  B )
32sseld 3011 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
41, 3mpd 13 1  |-  ( ph  ->  C  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1436    C_ wss 2986
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-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-in 2992  df-ss 2999
This theorem is referenced by:  exmid01  3999  frirrg  4144  ordtri2or2exmid  4353  riotass  5577  tfrcldm  6063  eroveu  6316  eroprf  6318  findcard2d  6540  findcard2sd  6541  undifdc  6564  fisseneq  6570  suplub2ti  6617  nnppipi  6823  archnqq  6897  prarloclemlt  6973  suprubex  8324  suprzclex  8754  fzssp1  9389  elfzoelz  9462  fzofzp1  9541  fzostep1  9551  frecuzrdgg  9726  frecuzrdgdomlem  9727  frecuzrdgsuctlem  9733  iseqvalt  9765  isermono  9786  bcm1k  10017  fimaxre2  10497  zssinfcl  10738
  Copyright terms: Public domain W3C validator