Theorem sseldi 2998

Description: Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Hypotheses

Ref	Expression
sseli.1	|- A C_ B
sseldi.2	|- ( ph -> C e. A )

Assertion

Ref	Expression
sseldi	|- ( ph -> C e. B )

Proof of Theorem sseldi

Step	Hyp	Ref	Expression
1		sseldi.2	. 2 |- ( ph -> C e. A )
2		sseli.1	. . 3 |- A C_ B
3	2	sseli 2996	. 2 |- ( C e. A -> C e. B )
4	1, 3	syl 14	1 |- ( ph -> C e. B )

Syntax hints:   -> wi 4   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:  riotacl  5507  riotasbc  5508  elmpt2cl  5723  ofrval  5747  f1od2  5881  mpt2xopn0yelv  5882  tpostpos  5907  smores  5935  supubti  6461  suplubti  6462  prarloclemcalc  6743  rereceu  7106  recriota  7107  rexrd  7219  nnred  8108  nncnd  8109  un0addcl  8377  un0mulcl  8378  nnnn0d  8397  nn0red  8398  nn0xnn0d  8416  suprzclex  8515  nn0zd  8537  zred  8539  rpred  8843  ige2m1fz  9192  zmodfzp1  9419  iseqcaopr2  9546  expcl2lemap  9574  m1expcl  9585  lcmn0cl  10583