Theorem sselii 3181

Description: Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseli.1	⊢ 𝐴 ⊆ 𝐵
sselii.2	⊢ 𝐶 ∈ 𝐴

Assertion

Ref	Expression
sselii	⊢ 𝐶 ∈ 𝐵

Proof of Theorem sselii

Step	Hyp	Ref	Expression
1		sselii.2	. 2 ⊢ 𝐶 ∈ 𝐴
2		sseli.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
3	2	sseli 3180	. 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)
4	1, 3	ax-mp 5	1 ⊢ 𝐶 ∈ 𝐵

Syntax hints:   ∈ wcel 2167   ⊆ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  brtpos0  6319  ax1cn  7945  recni  8055  0xr  8090  pnfxr  8096  nn0rei  9277  0xnn0  9335  nnzi  9364  nn0zi  9365  mincncf  14936  lgsdir2lem3  15355