Theorem sseldi 2937

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 2935	. 2 |- ( C e. A -> C e. B)
4	1, 3	syl 14	1 |- (ph -> C e. B)

Syntax hints:   -> wi 4   e. wcel 1390   C_ wss 2911

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  riotacl  5425  riotasbc  5426  elmpt2cl  5640  ofrval  5664  mpt2xopn0yelv  5795  tpostpos  5820  smores  5848  prarloclemcalc  6485  rexrd  6872  nnred  7708  nncnd  7709  un0addcl  7991  un0mulcl  7992  nnnn0d  8011  nn0red  8012  nn0zd  8134  zred  8136  rpred  8397  ige2m1fz  8742  expcl2lemap  8921  m1expcl  8932