Theorem bnj1293 35213

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 4189	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐶
3	1, 2	eqsstri 3982	1 ⊢ 𝐴 ⊆ 𝐶

This proof depends on syntax axioms:   = wceq 1569   ∩ cin 3903   ⊆ wss 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921

This theorem is used by:  bnj1253  35414  bnj1286  35416  bnj1280  35417  bnj1296  35418