ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  difprsn1 Unicode version

Theorem difprsn1 3629
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 2369 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 disjsn2 3556 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
3 disj3 3385 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
42, 3sylib 121 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
5 df-pr 3504 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
65equncomi 3192 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
76difeq1i 3160 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
8 difun2 3412 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
97, 8eqtri 2138 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
104, 9syl6reqr 2169 . 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 1316    =/= wne 2285    \ cdif 3038    u. cun 3039    i^i cin 3040   (/)c0 3333   {csn 3497   {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504
This theorem is referenced by:  difprsn2  3630
  Copyright terms: Public domain W3C validator