Theorem ssinss1 3351

Description: Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Assertion

Ref	Expression
ssinss1	⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem ssinss1

Step	Hyp	Ref	Expression
1		inss1 3342	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		sstr2 3149	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 3115   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by:  inss  3352  distop  12725  ntrin  12764  innei  12803  txcnp  12911