Theorem inabs 4220

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 4132	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		dfss2 3924	. 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴)
3	1, 2	mpbi 233	1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

Syntax hints:   = wceq 1570   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-in 3913  df-ss 3923

This theorem is referenced by:  dfif5  4505  caragenuncllem  47206