Theorem ssinss1OLD 4198

Description: Obsolete version of ssinss1 4197 as of 10-Jun-2026. (Contributed by NM, 14-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ssinss1OLD	⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

Proof of Theorem ssinss1OLD

Step	Hyp	Ref	Expression
1		inss1 4188	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		sstr2 3943	. 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 3903   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-ss 3921

This theorem is referenced by: (None)