Theorem sselda 2999

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

Step	Hyp	Ref	Expression
1		sseld.1	. . 3 |- ( ph -> A C_ B )
2	1	sseld 2998	. 2 |- ( ph -> ( C e. A -> C e. B ) )
3	2	imp 122	1 |- ( ( ph /\ C e. A ) -> C e. B )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   C_ wss 2973

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  elrel  4460  ffvresb  5349  1stdm  5828  tfrlem1  5946  tfrlemiubacc  5967  erinxp  6203  fundmen  6309  supisolem  6421  ordiso2  6446  elprnql  6671  elprnqu  6672  suprleubex  8032  un0addcl  8321  un0mulcl  8322  suprzclex  8445  supminfex  8685  icoshftf1o  9013  elfzom1elfzo  9212  zpnn0elfzo  9216  iseqfveq  9450  monoord2  9456  rexanre  10106  rexico  10107