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Theorem undifabs 3327
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 2991 . . 3 𝐴𝐴
2 difss 3097 . . 3 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3145 . 2 (𝐴 ∪ (𝐴𝐵)) ⊆ 𝐴
4 ssun1 3133 . 2 𝐴 ⊆ (𝐴 ∪ (𝐴𝐵))
53, 4eqssi 2988 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cdif 2941  cun 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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