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Theorem inundifss 3512
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 3367 . 2 (𝐴𝐵) ⊆ 𝐴
2 difss 3273 . 2 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3322 1 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cdif 3138  cun 3139  cin 3140  wss 3141
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154
This theorem is referenced by:  resasplitss  5407
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