Theorem ssinss1 3194

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 3186	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		sstr2 3006	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶))
3	1, 2	ax-mp 7	1 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Syntax hints:   → wi 4   ∩ cin 2972   ⊆ 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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  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-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by:  inss  3195