Theorem bnj1292 30140

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 3795	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
3	1, 2	eqsstri 3598	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1475   ∩ cin 3539   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554

This theorem is referenced by:  bnj1253  30339  bnj1286  30341  bnj1280  30342  bnj1296  30343