Theorem inabs 3487

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 3427	. 2 ⊢ A ⊆ (A ∪ B)
2		df-ss 3260	. 2 ⊢ (A ⊆ (A ∪ B) ↔ (A ∩ (A ∪ B)) = A)
3	1, 2	mpbi 199	1 ⊢ (A ∩ (A ∪ B)) = A

Syntax hints:   = wceq 1642   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  dfif5  3675