Theorem unabs 4200

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 4172	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		ssequn2 4125	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴)
3	1, 2	mpbi 231	1 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴

Syntax hints:   = wceq 1547   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-in 3897  df-ss 3907

This theorem is referenced by:  volun  25537