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Theorem difprsn1 3544
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 2333 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 disjsn2 3473 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
3 disj3 3312 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
42, 3sylib 120 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
5 df-pr 3423 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
65equncomi 3128 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
76difeq1i 3096 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
8 difun2 3338 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
97, 8eqtri 2103 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
104, 9syl6reqr 2134 . 2  |-  ( B  =/=  A  ->  ( { A ,  B }  \  { A } )  =  { B }
)
111, 10sylbir 133 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    =/= wne 2249    \ cdif 2979    u. cun 2980    i^i cin 2981   (/)c0 3267   {csn 3416   {cpr 3417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423
This theorem is referenced by:  difprsn2  3545
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