Theorem sseld 2944

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 2939	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ 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:  sselda  2945  sseldd  2946  ssneld  2947  elelpwi  3370  ssbrd  3805  uniopel  3993  onintonm  4243  sucprcreg  4273  ordsuc  4287  0elnn  4340  dmrnssfld  4595  nfunv  4933  opelf  5062  fvimacnv  5282  ffvelrn  5300  f1imass  5413  dftpos3  5877  nnmordi  6089  diffifi  6351  ordiso2  6357  prarloclemarch2  6517  ltexprlemrl  6708  cauappcvgprlemladdrl  6755  caucvgprlemladdrl  6776  caucvgprlem1  6777  uzind  8349  ixxssxr  8769  elfz0add  8979  elfz0addOLD  8980  fzoss1  9027  iseqss  9226  bj-nnord  10083