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Theorem difprsn1 3927
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 2679 . 2  |-  ( B  =/=  A  <->  A  =/=  B )
2 disjsn2 3861 . . . 4  |-  ( B  =/=  A  ->  ( { B }  i^i  { A } )  =  (/) )
3 disj3 3664 . . . 4  |-  ( ( { B }  i^i  { A } )  =  (/) 
<->  { B }  =  ( { B }  \  { A } ) )
42, 3sylib 189 . . 3  |-  ( B  =/=  A  ->  { B }  =  ( { B }  \  { A } ) )
5 df-pr 3813 . . . . . 6  |-  { A ,  B }  =  ( { A }  u.  { B } )
65equncomi 3485 . . . . 5  |-  { A ,  B }  =  ( { B }  u.  { A } )
76difeq1i 3453 . . . 4  |-  ( { A ,  B }  \  { A } )  =  ( ( { B }  u.  { A } )  \  { A } )
8 difun2 3699 . . . 4  |-  ( ( { B }  u.  { A } )  \  { A } )  =  ( { B }  \  { A } )
97, 8eqtri 2455 . . 3  |-  ( { A ,  B }  \  { A } )  =  ( { B }  \  { A }
)
104, 9syl6reqr 2486 . 2  |-  ( B  =/=  A  ->  ( { A ,  B }  \  { A } )  =  { B }
)
111, 10sylbir 205 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    =/= wne 2598    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   {cpr 3807
This theorem is referenced by:  difprsn2  3928  usgra1v  21401  cusgra2v  21463  coinflippvt  24734  f12dfv  28066  frgra2v  28326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813
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