Theorem bnj1293 33642

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 4222	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐶
3	1, 2	eqsstri 4009	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1541   ∩ cin 3940   ⊆ wss 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3430  df-v 3472  df-in 3948  df-ss 3958

This theorem is referenced by:  bnj1253  33843  bnj1286  33845  bnj1280  33846  bnj1296  33847