Theorem bnj1292 34125

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1292.1	⊢ 𝐴 = (𝐵 ∩ 𝐶)

Assertion

Ref	Expression
bnj1292	⊢ 𝐴 ⊆ 𝐵

Proof of Theorem bnj1292

Step	Hyp	Ref	Expression
1		bnj1292.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
2		inss1 4228	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
3	1, 2	eqsstri 4016	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1540   ∩ cin 3947   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965

This theorem is referenced by:  bnj1253  34327  bnj1286  34329  bnj1280  34330  bnj1296  34331