Theorem inabs 4189

Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Assertion

Ref	Expression
inabs	⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

Proof of Theorem inabs

Step	Hyp	Ref	Expression
1		ssun1 4106	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		df-ss 3904	. 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

Syntax hints:   = wceq 1539   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904

This theorem is referenced by:  dfif5  4475  caragenuncllem  44050