Theorem sselda 3000

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 2999	. 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 1434   C_ wss 2974

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987

This theorem is referenced by:  elrel  4468  ffvresb  5360  1stdm  5839  tfrlem1  5957  tfrlemiubacc  5979  tfr1onlemubacc  5995  tfrcllemubacc  6008  erinxp  6246  fundmen  6353  supisolem  6480  ordiso2  6505  elprnql  6733  elprnqu  6734  suprleubex  8099  un0addcl  8388  un0mulcl  8389  suprzclex  8526  supminfex  8766  icoshftf1o  9089  elfzom1elfzo  9289  zpnn0elfzo  9293  iseqfveq  9546  monoord2  9552  rexanre  10244  rexico  10245