Theorem unabs 4199

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 4173	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		ssequn2 4128	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴

Syntax hints:   = wceq 1540   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-un 3902  df-in 3904  df-ss 3914

This theorem is referenced by:  volun  24781