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Theorem difprsn1 3730
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 2431 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 df-pr 3598 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
32equncomi 3281 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
43difeq1i 3249 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
5 difun2 3502 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
64, 5eqtri 2198 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
7 disjsn2 3654 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
8 disj3 3475 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
97, 8sylib 122 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
106, 9eqtr4id 2229 . 2  |-  ( B  =/=  A  ->  ( { A ,  B }  \  { A } )  =  { B }
)
111, 10sylbir 135 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    =/= wne 2347    \ cdif 3126    u. cun 3127    i^i cin 3128   (/)c0 3422   {csn 3591   {cpr 3592
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598
This theorem is referenced by:  difprsn2  3731
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