Theorem unabs 4264

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

Assertion

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

Proof of Theorem unabs

Step	Hyp	Ref	Expression
1		inss1 4238	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		ssequn2 4192	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴

Syntax hints:   = wceq 1534   ∪ cun 3956   ∩ cin 3957   ⊆ wss 3958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-v 3474  df-un 3963  df-in 3965  df-ss 3975

This theorem is referenced by:  volun  25565