Theorem bnj1292 32082

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 4204	. 2 ⊢ (𝐵 ∩ 𝐶) ⊆ 𝐵
3	1, 2	eqsstri 4000	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1533   ∩ cin 3934   ⊆ wss 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-ss 3951

This theorem is referenced by:  bnj1253  32284  bnj1286  32286  bnj1280  32287  bnj1296  32288