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Theorem undif1ss 3339
 Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif1ss ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)

Proof of Theorem undif1ss
StepHypRef Expression
1 difss 3110 . 2 (𝐴𝐵) ⊆ 𝐴
2 unss1 3153 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵))
31, 2ax-mp 7 1 ((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∖ cdif 2981   ∪ cun 2982   ⊆ wss 2984 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997 This theorem is referenced by:  undif2ss  3340  pwundifss  4076
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