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Theorem undifabs 3405
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem undifabs
StepHypRef Expression
1 ssid 3083 . . 3 𝐴𝐴
2 difss 3168 . . 3 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3217 . 2 (𝐴 ∪ (𝐴𝐵)) ⊆ 𝐴
4 ssun1 3205 . 2 𝐴 ⊆ (𝐴 ∪ (𝐴𝐵))
53, 4eqssi 3079 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1314  cdif 3034  cun 3035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050
This theorem is referenced by:  exmid1stab  12887
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