Theorem unabs 4081

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 4053	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		ssequn2 4009	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴)
3	1, 2	mpbi 222	1 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴

Syntax hints:   = wceq 1601   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-un 3797  df-in 3799  df-ss 3806

This theorem is referenced by:  volun  23753