Theorem inabs 3888

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 3809	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		df-ss 3621	. 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴)
3	1, 2	mpbi 220	1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

Syntax hints:   = wceq 1523   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621

This theorem is referenced by:  dfif5  4135  caragenuncllem  41047