Theorem sseldi 2943

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 2941	. 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 2917

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 2924  df-ss 2931

This theorem is referenced by:  riotacl  5482  riotasbc  5483  elmpt2cl  5698  ofrval  5722  mpt2xopn0yelv  5854  tpostpos  5879  smores  5907  prarloclemcalc  6600  rereceu  6963  recriota  6964  rexrd  7075  nnred  7927  nncnd  7928  un0addcl  8215  un0mulcl  8216  nnnn0d  8235  nn0red  8236  nn0zd  8358  zred  8360  rpred  8622  ige2m1fz  8972  iseqcaopr2  9241  expcl2lemap  9267  m1expcl  9278