Theorem sseldi 2998

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 2996	. 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
4	1, 3	syl 14	1 ⊢ (𝜑 → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1434   ⊆ 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:  riotacl  5513  riotasbc  5514  elmpt2cl  5729  ofrval  5753  f1od2  5887  mpt2xopn0yelv  5888  tpostpos  5913  smores  5941  supubti  6471  suplubti  6472  prarloclemcalc  6754  rereceu  7117  recriota  7118  rexrd  7230  nnred  8119  nncnd  8120  un0addcl  8388  un0mulcl  8389  nnnn0d  8408  nn0red  8409  nn0xnn0d  8427  suprzclex  8526  nn0zd  8548  zred  8550  rpred  8854  ige2m1fz  9203  zmodfzp1  9430  iseqcaopr2  9557  expcl2lemap  9585  m1expcl  9596  lcmn0cl  10594