Theorem bnj1293 35013

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 4169	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐶
3	1, 2	eqsstri 3963	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1548   ∩ cin 3884   ⊆ wss 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-in 3892  df-ss 3902

This theorem is referenced by:  bnj1253  35214  bnj1286  35216  bnj1280  35217  bnj1296  35218