Theorem sseldi 2940

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 2938	. 2 |- ( C e. A -> C e. B )
4	1, 3	syl 14	1 |- ( ph -> C e. B )

Syntax hints:   -> wi 4   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:  riotacl  5469  riotasbc  5470  elmpt2cl  5685  ofrval  5709  mpt2xopn0yelv  5841  tpostpos  5866  smores  5894  prarloclemcalc  6581  rereceu  6944  recriota  6945  rexrd  7055  nnred  7903  nncnd  7904  un0addcl  8187  un0mulcl  8188  nnnn0d  8207  nn0red  8208  nn0zd  8330  zred  8332  rpred  8593  ige2m1fz  8939  iseqcaopr2  9119  expcl2lemap  9145  m1expcl  9156