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Theorem difprsn1 4152
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 2723 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 disjsn2 4077 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
3 disj3 3859 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
42, 3sylib 196 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
5 df-pr 4019 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
65equncomi 3636 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
76difeq1i 3604 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
8 difun2 3895 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
97, 8eqtri 2483 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
104, 9syl6reqr 2514 . 2  |-  ( B  =/=  A  ->  ( { A ,  B }  \  { A } )  =  { B }
)
111, 10sylbir 213 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    =/= wne 2649    \ cdif 3458    u. cun 3459    i^i cin 3460   (/)c0 3783   {csn 4016   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by:  difprsn2  4153  f12dfv  6154  pmtrprfval  16711  usgra1v  24592  cusgra2v  24664  frgra2v  25201  eulerpartlemgf  28582  coinflippvt  28687  ldepsnlinc  33363
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