Theorem undifabs 3294

Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Assertion

Ref	Expression
undifabs	|- (A u. (A \ B)) = A

Proof of Theorem undifabs

Step	Hyp	Ref	Expression
1		ssid 2958	. . 3 |- A C_ A
2		difss 3064	. . 3 |- (A \ B) C_ A
3	1, 2	unssi 3112	. 2 |- (A u. (A \ B)) C_ A
4		ssun1 3100	. 2 |- A C_ (A u. (A \ B))
5	3, 4	eqssi 2955	1 |- (A u. (A \ B)) = A

Syntax hints:   = wceq 1242   \ cdif 2908   u. cun 2909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925

This theorem is referenced by: (None)