Theorem sseld 2999

Description: Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Hypothesis

Ref	Expression
sseld.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem sseld

Step	Hyp	Ref	Expression
1		sseld.1	. 2 |- ( ph -> A C_ B )
2		ssel 2994	. 2 |- ( A C_ B -> ( C e. A -> C e. B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C e. A -> 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:  sselda  3000  sseldd  3001  ssneld  3002  elelpwi  3401  ssbrd  3834  uniopel  4019  onintonm  4269  sucprcreg  4300  ordsuc  4314  0elnn  4366  dmrnssfld  4623  nfunv  4963  opelf  5093  fvimacnv  5314  ffvelrn  5332  f1imass  5445  dftpos3  5911  nnmordi  6155  diffifi  6428  ordiso2  6505  prarloclemarch2  6671  ltexprlemrl  6862  cauappcvgprlemladdrl  6909  caucvgprlemladdrl  6930  caucvgprlem1  6931  uzind  8539  supinfneg  8764  infsupneg  8765  ixxssxr  8999  elfz0add  9211  fzoss1  9257  frecuzrdgrclt  9497  iseqss  9541  fisumcvg  10338  bj-nnord  10911