Theorem undifabs 3538

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 3214	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		difss 3300	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	1, 2	unssi 3349	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
4		ssun1 3337	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵))
5	3, 4	eqssi 3210	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:   = wceq 1373   ∖ cdif 3164   ∪ cun 3165

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180

This theorem is referenced by:  exmid1stab  4256