Theorem bnj1292 34788

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 4217	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
3	1, 2	eqsstri 4010	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1539   ∩ cin 3930   ⊆ wss 3931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-in 3938  df-ss 3948

This theorem is referenced by:  bnj1253  34990  bnj1286  34992  bnj1280  34993  bnj1296  34994