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Theorem undif1ss 3432
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 3197 . 2  |-  ( A 
\  B )  C_  A
2 unss1 3240 . 2  |-  ( ( A  \  B ) 
C_  A  ->  (
( A  \  B
)  u.  B ) 
C_  ( A  u.  B ) )
31, 2ax-mp 5 1  |-  ( ( A  \  B )  u.  B )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 3063    u. cun 3064    C_ wss 3066
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079
This theorem is referenced by:  undif2ss  3433  pwundifss  4202
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