ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difprsn2 Unicode version

Theorem difprsn2 3551
Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
difprsn2  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)

Proof of Theorem difprsn2
StepHypRef Expression
1 prcom 3492 . . 3  |-  { A ,  B }  =  { B ,  A }
21difeq1i 3098 . 2  |-  ( { A ,  B }  \  { B } )  =  ( { B ,  A }  \  { B } )
3 necom 2333 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 difprsn1 3550 . . 3  |-  ( B  =/=  A  ->  ( { B ,  A }  \  { B } )  =  { A }
)
53, 4sylbi 119 . 2  |-  ( A  =/=  B  ->  ( { B ,  A }  \  { B } )  =  { A }
)
62, 5syl5eq 2127 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    =/= wne 2249    \ cdif 2981   {csn 3422   {cpr 3423
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 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator