Theorem sseldi 2997

Description: Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Hypotheses

Ref	Expression
sseli.1	⊢ 𝐴 ⊆ 𝐵
sseldi.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
sseldi	⊢ (𝜑 → 𝐶 ∈ 𝐵)

Proof of Theorem sseldi

Step	Hyp	Ref	Expression
1		sseldi.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
2		sseli.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
3	2	sseli 2995	. 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
4	1, 3	syl 14	1 ⊢ (𝜑 → 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:  riotacl  5502  riotasbc  5503  elmpt2cl  5718  ofrval  5742  f1od2  5876  mpt2xopn0yelv  5877  tpostpos  5902  smores  5930  supubti  6412  suplubti  6413  prarloclemcalc  6692  rereceu  7055  recriota  7056  rexrd  7168  nnred  8052  nncnd  8053  un0addcl  8321  un0mulcl  8322  nnnn0d  8341  nn0red  8342  suprzclex  8445  nn0zd  8467  zred  8469  rpred  8773  ige2m1fz  9127  zmodfzp1  9350  iseqcaopr2  9461  expcl2lemap  9488  m1expcl  9499  lcmn0cl  10450