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Theorem undif1ss 3298
Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif1ss  |-  ( ( A  \  B )  u.  B )  C_  ( A  u.  B
)

Proof of Theorem undif1ss
StepHypRef Expression
1 difss 3070 . 2  |-  ( A 
\  B )  C_  A
2 unss1 3112 . 2  |-  ( ( A  \  B ) 
C_  A  ->  (
( A  \  B
)  u.  B ) 
C_  ( A  u.  B ) )
31, 2ax-mp 7 1  |-  ( ( A  \  B )  u.  B )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 2914    u. cun 2915    C_ wss 2917
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 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931
This theorem is referenced by:  undif2ss  3299  pwundifss  4022
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