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Theorem undif2ss 3377
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 3376 . 2  |-  ( ( B  \  A )  u.  A )  C_  ( B  u.  A
)
2 uncom 3159 . 2  |-  ( A  u.  ( B  \  A ) )  =  ( ( B  \  A )  u.  A
)
3 uncom 3159 . 2  |-  ( A  u.  B )  =  ( B  u.  A
)
41, 2, 33sstr4i 3080 1  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 3010    u. cun 3011    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: (None)
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