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Theorem inundifss 3408
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 3264 . 2 (𝐴𝐵) ⊆ 𝐴
2 difss 3170 . 2 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3219 1 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cdif 3036  cun 3037  cin 3038  wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052
This theorem is referenced by:  resasplitss  5270
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