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Theorem undif2ss 3510
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 3509 . 2  |-  ( ( B  \  A )  u.  A )  C_  ( B  u.  A
)
2 uncom 3291 . 2  |-  ( A  u.  ( B  \  A ) )  =  ( ( B  \  A )  u.  A
)
3 uncom 3291 . 2  |-  ( A  u.  B )  =  ( B  u.  A
)
41, 2, 33sstr4i 3208 1  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 3138    u. cun 3139    C_ 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: (None)
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