Theorem undifabs 3568

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

Assertion

Ref	Expression
undifabs	⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Proof of Theorem undifabs

Step	Hyp	Ref	Expression
1		ssid 3244	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		difss 3330	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	1, 2	unssi 3379	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
4		ssun1 3367	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵))
5	3, 4	eqssi 3240	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:   = wceq 1395   ∖ cdif 3194   ∪ cun 3195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  exmid1stab  4291