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 3920	. 2 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴)
3	1, 2	mpbi 233	1 ⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-in 3909  df-ss 3919

This theorem is used by:  dfif5  4502  caragenuncllem  47348