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Theorem inundifss 3379
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  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  C_  A

Proof of Theorem inundifss
StepHypRef Expression
1 inss1 3235 . 2  |-  ( A  i^i  B )  C_  A
2 difss 3141 . 2  |-  ( A 
\  B )  C_  A
31, 2unssi 3190 1  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  C_  A
Colors of variables: wff set class
Syntax hints:    \ cdif 3010    u. cun 3011    i^i cin 3012    C_ wss 3013
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026
This theorem is referenced by:  resasplitss  5225
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