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Theorem inundifss 3515
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 3370 . 2 (𝐴𝐵) ⊆ 𝐴
2 difss 3276 . 2 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3325 1 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cdif 3141  cun 3142  cin 3143  wss 3144
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by:  resasplitss  5410
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