Theorem sseld 2998

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

Hypothesis

Ref	Expression
sseld.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
sseld	⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))

Proof of Theorem sseld

Step	Hyp	Ref	Expression
1		sseld.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssel 2993	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   ∈ wcel 1433   ⊆ wss 2973

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  sselda  2999  sseldd  3000  ssneld  3001  elelpwi  3393  ssbrd  3826  uniopel  4011  onintonm  4261  sucprcreg  4292  ordsuc  4306  0elnn  4358  dmrnssfld  4613  nfunv  4953  opelf  5082  fvimacnv  5303  ffvelrn  5321  f1imass  5434  dftpos3  5900  nnmordi  6112  diffifi  6378  ordiso2  6446  prarloclemarch2  6609  ltexprlemrl  6800  cauappcvgprlemladdrl  6847  caucvgprlemladdrl  6868  caucvgprlem1  6869  uzind  8458  supinfneg  8683  infsupneg  8684  ixxssxr  8923  elfz0add  9134  fzoss1  9180  iseqss  9446  bj-nnord  10753