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Theorem inundifss 3364
Description: The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
inundifss ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴

Proof of Theorem inundifss
StepHypRef Expression
1 inss1 3221 . 2 (𝐴𝐵) ⊆ 𝐴
2 difss 3127 . 2 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3176 1 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cdif 2997  cun 2998  cin 2999  wss 3000
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013
This theorem is referenced by:  resasplitss  5203
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