Theorem bnj1292 35311

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 4185	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
3	1, 2	eqsstri 3980	1 ⊢ 𝐴 ⊆ 𝐵

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3901   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-ss 3919

This theorem is used by:  bnj1253  35513  bnj1286  35515  bnj1280  35516  bnj1296  35517