Theorem bnj1293 34792

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

Hypothesis

Ref	Expression
bnj1293.1	⊢ 𝐴 = (𝐵 ∩ 𝐶)

Assertion

Ref	Expression
bnj1293	⊢ 𝐴 ⊆ 𝐶

Proof of Theorem bnj1293

Step	Hyp	Ref	Expression
1		bnj1293.1	. 2 ⊢ 𝐴 = (𝐵 ∩ 𝐶)
2		inss2 4259	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐶
3	1, 2	eqsstri 4043	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1537   ∩ cin 3975   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993

This theorem is referenced by:  bnj1253  34993  bnj1286  34995  bnj1280  34996  bnj1296  34997