Theorem inabs 4215

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 4127	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		dfss2 3916	. 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴)
3	1, 2	mpbi 230	1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

Syntax hints:   = wceq 1541   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-in 3905  df-ss 3915

This theorem is referenced by:  dfif5  4491  caragenuncllem  46634