Theorem undifabs 3573

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 3248	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		difss 3335	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	1, 2	unssi 3384	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
4		ssun1 3372	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵))
5	3, 4	eqssi 3244	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:   = wceq 1398   ∖ cdif 3198   ∪ cun 3199

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214

This theorem is referenced by:  exmid1stab  4304