Theorem undifabs 3501

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 3177	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		difss 3263	. . 3 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	1, 2	unssi 3312	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
4		ssun1 3300	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (𝐴 ∖ 𝐵))
5	3, 4	eqssi 3173	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:   = wceq 1353   ∖ cdif 3128   ∪ cun 3129

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144

This theorem is referenced by:  exmid1stab  4210