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Theorem inundifss 3298
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 3154 . 2 (𝐴𝐵) ⊆ 𝐴
2 difss 3067 . 2 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3115 1 ((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cdif 2911  cun 2912  cin 2913  wss 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928
This theorem is referenced by:  resasplitss  5032
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