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Theorem undif2ss 3469
Description: Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif2ss  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)

Proof of Theorem undif2ss
StepHypRef Expression
1 undif1ss 3468 . 2  |-  ( ( B  \  A )  u.  A )  C_  ( B  u.  A
)
2 uncom 3251 . 2  |-  ( A  u.  ( B  \  A ) )  =  ( ( B  \  A )  u.  A
)
3 uncom 3251 . 2  |-  ( A  u.  B )  =  ( B  u.  A
)
41, 2, 33sstr4i 3169 1  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 3099    u. cun 3100    C_ wss 3102
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by: (None)
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