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Theorem difprsn1 3663
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
difprsn1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)

Proof of Theorem difprsn1
StepHypRef Expression
1 necom 2393 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 df-pr 3535 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
32equncomi 3223 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
43difeq1i 3191 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
5 difun2 3443 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
64, 5eqtri 2161 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
7 disjsn2 3590 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
8 disj3 3416 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
97, 8sylib 121 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
106, 9eqtr4id 2192 . 2  |-  ( B  =/=  A  ->  ( { A ,  B }  \  { A } )  =  { B }
)
111, 10sylbir 134 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    =/= wne 2309    \ cdif 3069    u. cun 3070    i^i cin 3071   (/)c0 3364   {csn 3528   {cpr 3529
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rab 2426  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-sn 3534  df-pr 3535
This theorem is referenced by:  difprsn2  3664
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